Theoretical Basis: Plasma Focus Model (Radiative)-S Lee Model

Transcription

1 Theeil Bi: Plm Fu Mel (Riive)-S Lee Mel h: (Thi eviin, 7 Mh 008, nm RADPFV5.3.8, inluing bem-ge neun yiel n lm el-bin wih mh niin m vlumei ue emiin) Thi mel h been evele Mhe-ye () lm u mhine. w evele he 3kJ mhine knwn he UNUCTP PFF (,3) (Unie Nin Univeiyneninl Cene Theeil Phyi Plm Fu iliy, whih nw m n ineninl newk.n inil hee i n limi enegy ge n elee niguin, hugh hue-keeing my nee be ie u in exeme e, in e kee wihin eiien nge e.g. gh ling. F eil he muing kge, g bk he inuy ein. h: The mel h been ue viu liin, exmle, in he eign ing lm u (Re 4); n eiming x-y yiel (Re 5) he ue eveling SXR ue mieleni lihghy (Re 6); n eenly in unveing inh uen limiin ee (Re 7, 008), hwing new ligh n neun ling lw (Re 8, 008) n n exeimenl ehnique (Re 9, 008) mue u inh uen m meue ihge uen wvem. The 5-he mel i eibe in me eil in he llwing ein: Axil Phe Ril nw Shk Phe 3 Ril Relee Shk Phe 4 Slw Cmein (Riive) Phe 5 Exne Clumn Axil Phe

2 O b Axil Phe (nw-lw mel) Oue elee nne elee 0 Fig () Fig (b) Re hnge mmenum uen heh, iin, i mv) [ ρ ( b ) ] ρ ( ) ( m Mgnei e n uen heh i m F b () ( ) 4 m in m we wn he ube in he xil iein in uen lwing in in Equin min: ρ ( ) m ( ) 4 m ( ) 4 ρ ( ) -- ()

3 Ciui (uen) Equin L C V L() Fig Ciui hemi S () gne (), lm eine. Thi i he ximin whih i genelly ue elemgnei ive. ( L [( L L ) ] L ) V V C C L V C ( ) L ( ) -- () Equin () n ) e he geneing equin he mel. They nin he hyi buil in he mel. They e ule equin. The equin min i ee by he elei uen. The iui equin i ee by he uen heh min n iin. Nmlie he equin bin ling mee Rele vible,, by nn-imeninlie quniie llw:, ς, ι whee he nmliing quniie, n Z e eully hen be elevn, heii, nvenien quniie, eleing he hyi he blem. Chie: i he lengh he ne, i L C (ning h L C i he yle ime L -C ihge iui) 3

4 4 i Z V whee C L Z i he uge imene (ning h i he ek uen he L -C ihge iui wih i C hge iniilly V.) Nmliing, we hve: Equin min: ( ) ι ρ ι ρ m m ) ( 4 4 whih we wie ι α -- (.) Obin i ling mee: We ne, by inein, ( ) [ ]( ) ( ) [ ] { } 4 m ρ α, whih we hu eine in hi mnne. By inein equin (.), we ne α i imeninle. Hene ine h he imenin ime we my eine ime vlue whee ( ) ( ) ρ 4 m ieniying hi quniy he heii xil ni ime he CS wn he ne xil he. We my hen hink α : α ( ) ling mee. i heii eleil ihge ime heii xil ni ime. We my uhe ieniy heii xil ni ee V ( ) ( ) ρ v 4 m

5 The quniy u, inhe e. ρ i he S (ee ive) elemgneilly iven evie, Nmliing he iui (uen) Equin, we hve: ι v ι ι ( ) ι L ( ) n ubiuing in V L C, L C, we hve ι ι ) ( ) L ( Z ι ( ) L ι wie: ι βι δι ( β ) -- (.) Sen ling mee We ne L ( ) i he inune he xil he when CS ehe he en. L Hene β i he i l ue inune n ine he evie i elemgnei, he L eleynmi i eemine ngly by hi ling mee. The hi ling mee δ Z i he i iui y eine uge imene. Thi ming ee n he uen. (.) n (.) e he Geneing Equin h my be inege e-by-e. Clule vlge inu eminl u ube: V L ( L ) L whee L ( ) - (.0) V Nmlie m υ βι β - (.) V negin Deine iniil niin: ι 0, 0, 0, ι 0, ι 0,, α 3 Se ime inemen: D

6 nemen ime: D Nex e vlue e mue uing he llwing line ximin: D D ι ι ι D ι ι D ι Ue new vlue,, ι n ι lule new geneing vlue geneing equ (.) n (.). ι n uing nemen ime gin n ee lulin nex e vlue n new geneing vlue. Cninue eue unil. Then g n il he inw hk muin. Ril nw Shk Phe (Slug mel) Cmuin & exeimen Ril inw hk he RS he exeimen Pinh iemble Exne lumn he iu m min Fig 3. Shemi il he The nwlw mel i ue xil he ju bin xil jey n ee (m whih emeue my be eue) n bin enble uen ile. A he CS i ume be ininiemlly hin, n inmin eniy i nine in he hyi he equin min, lhugh n eime eniy my be bine by invking iinl mehnim e.g. uing hk wve hey. n he il he hweve, nwlw mel (wih n ininiemlly hin CS) wul evenully (in he inegin) le ll uen lwing 0, wih ininie inune n eniy. We hu ele he nw lw mel by lug mel. n hi mel, he mgnei eue ive hk wve he i, eing e he mgnei in (CS) mve in. The ee he inw il hk n (ee Fig b)i eemine by he mgnei eue (whih een n he ive uen vlue n CS iin ). 6

7 The ee he mgnei in (CS) i eemine by he i lw hemynmi lie he eeive inee in vlume beween SF n CS, ee by he inemenl min he SF. The mein i ee n eging inh. Fu geneing equin e neee eibe he min () SF, (b) CS () inh egin n he elei uen (); inege he u vible,, &. Min Shk Fn: in SF P m P ρ, P, T vuum ρ T Fm Shk Wve hey, hk eue P ρ v whee hk ee v in mbien g ρ ue he eue he hke g (ju behin he hk n) ie vlue P. we ume h hi eue i unim m he SF he CS (ininie ui [mll iubne] ee ximin) hen he in, we my ly P P m whee P m ( ) Thu: v ( ) x 8 ρ m whee i he iui uen n i he uen lwing in he ylinil CS, ken he me in he xil he, n ρ m i he eeive m eniy we in he il lug; whee m i ieen (genelly lge) hn m he xil he. Thu ( ) -- () ρ 4 m Egin ee CS (en-ene bh en) The il mein i en ne en. Hene n xil hk i ge in he -iein, w he wnem ne xi. We ke he iin he xil CS (he hn he SF). The eue iving he xil hk i he me he eue iving he inw il hk. Thu he xil hk ee i he me he il hk ee. The CS ee i lwe, m hk wve hey, by n xime (). Thu he xil egin ee he CS i: 7

8 -- (V) n hi melling we e he egin in vey xime hin, i ee n he meing lumn i elively eny. The min mehnim nlling he e he lm lumn i he il mein. The il CS (in) ee i hene ee wih me e llw: Ril in min We inquie: F n inemenl min,, he hk n, iving uen, wh i he elinhi beween lm lug eue P n lm lug vlume V? We ume n ibi elinhi (7) (ininie mll iubne ee whih we will ly ein ubequenly) ixe m g in he lug uing he inemenl min. We hve PV nn V v P P 0 whee lug eue P ~ v bu P v P v v ~ (ee ein n Min Shk Fn, bve) P P Nw lug vlume V ( ) n i igh V ( ) ( ) n e! Bu hee we ne h lhugh he min he in e n hnge he m g in he lug, he min he hk n,, e wee in n mun mbien g. Thi mun we in i equl he mbien g we hugh by he hk n in i min. Thi we-u g i mee by i ()( -) n will uy he inee in vlume V. The ul inee in vlume vilble he iginl m g in vlume V e n en inemen bu n eeive (eue) inemen (()). (Ne i eii he i 8

9 9 he lm e.g. 53 mi g, 75 mleul g; ngly iniing gn h vlue le e.g..5.) Thu, he me e ineein i: ( ) V Thu we hve: ( ) ( ) V V n ing gehe PP n VV we hve ( ) ( ) 0 Renging n uing he ubje we hve ) ( -- (V) whee we e emine il in iin il hk n iin xil in iin Ciui Equin uing il he The inune he u ube nw ni he ull inune he xil he n he inune he illy imling & eging lm inh. Thu ) ( b L whee bh n vy wih ime. Thu he iui (uen) equin i C V b b L ) (

10 0 Giving ) ( b L b C V -- (V) Geneing equin (), (V), (V), (V) m le e equin whih my be inege,, n. Nmliin F hi he he llwing nmliin i e., ι in xil he bu wih,, ie. ine e nmlie ne iu, ine ne lengh. Ae nmliin we hve: Ril hk ee αα ι -- (.) Axil lumn egin ee (bh en lumn eine by xil in) -- (V.) Ril in ee: ) )( ( ) ( ι ι -- (V.) uen: [ ] ( ) ( ) [ ] { } ) ( β β δι ι β ι β ι ι k (V.) whee he ling mee e β β(f), F n ( )( ) [ ] [ ] m ) (4 m F α Ne h whee we inee α, we my inee α whee i heii il ni ime. The ling mee αα my hen be ineee αα We ne h α he i heii xil ni heii il mein inw hk ni ime i eenilly gemeil i

11 [( ) 4 ] ~ 0 ( i F -~6 n -~ 3) F (ie. xil ni ime i heiilly 0 ime ge hn il hk ni) miie by he hemynmi em ( ) n he m we u i ( ). i xil il heii ime i yilly 40. We l ne m he ling mee αα h 4 m ( [ ( ) ] 0 ρ n heii ee inw hk eh u xi i: v [ ( ) ] ( ) 4 ρ m m m nluing ll 3, he The i heii il n xil ee i l eenilly gemeil ne, miie by ( )( ) hemynmi. i v v 4 wih vlue yilly.5. Ne h he il heii ee h he me eenene he xil ni ee n ive S. ( ) ρ Clule vlge V PF inu eminl A in he xil he, he vlge i ken hve nly inuive mnen. V (L ) b whee L ( ) b b V ( ) - (V.0) We my l wie in nmlie m υ VV (nmlie iniil i vlge V ) ν β β ι β ι - (V.) The geneing equin (.), (V.), (V.), (V.) my nw be inege uing he llwing iniil niin: he ime h xil he ene 0 (ken mll numbe uh vi numeil iiulie equin V.)

12 ι vlue uen he en he xil he. ι vlue lwe hge en xil he. The inegin (e-by-e) my ee wih he llwing lgihm: (king mlle ime inemen D ) Uing iniil vlue (bve),, n ι ι,, n e equenilly lule m geneing equin (.), (V.), (V.), (V.). D D Then equenilly uing line ximin: D ι ι ι D ι ι D ι ι Time i hen inemene by D, n he nex e vlue,,, e mue m (.), (V.), (V.) n (V.), llwe by line ximin,, ρ, ι n ι. The equene i eee e-by-e unil 0. Cein inie ui (mll iubne) ee. n he lug mel bve we ume h he eue exee by he mgnei in (uen, iin ) i innneuly el by he hk n (iin ). Likewie he hk ee i innneuly el by he in (CS). Thi umin iniinie mll iubne ee (SDS) i imlii in equin () n (V) ( in nmlie m (.) n (V.)). Sine he SDS i inie, hee i ully ime le Δ mmuniing beween he SF n CS. Thi mmuniin ely h be ine in he mel. Ohewie he PF, he muin will yiel high vlue CS n SF ee. Cnie he inn, SF i, CS, vlue uen i. SF ully eel he ee he uen n vlue bu vlue ely whih lwe ime (-Δ), wih he CS ely. Similly he in hink he SF ee i n bu ime (-Δ). ely T imlemen hi inie SDS ein we he llwing eue: Clule he SDS, ken he ui ee.

13 P SDS ρ R M DT DkT Mmi whee eii he i, M Mleule Weigh, R univel G nn 8 x 0 3 (S uni) m i m n, k Blmnn nn. D eue eiien DN () whee Z, hee, i he eeive hge he lm J Z α, umme ve ll iniin level J. DN iiin numbe, e.g. Deueium DN, whee gn DN. The emeue T my be mue he hke lm T M RD ( ) ( ) Clule he mmuniin ely ime ΔT ( )SDS n u gmme uing he Mi EXCEL VSUAL BASC, he e-by-e inegin i e w-by-w, eh e ening ne w. Thu he ΔT my be nvee Δ (w numbe) by uing Δ (w numbe) ΔT(imee inemen) hi Δ (w numbe being, ue, une n inege. The ein hen invlve lking bk he elevn w numbe ex he ee vlue ely, ely, ely. Thu in he ul numeil inegin, in equin (.), ι n e ele by ι ely n ely k n in equin (V.) k i ele by ely Ril Relee Shk Phe When he inw il hk hi he xi, 0. Thu in he muin, when we exi m il inw hk he. We muing he RS he. The RS i given nn ee 0.3 n-xi inw il hk ee. n hi he muin i ie u in el (S) uni. 3

14 Relee Shk See: 0. 3 nxi Pin ee: Ue he me equin V exe u 0 n 0 Egin ee: Ue me equin Eq V. nxi Ciui Equin: Ue he me equin Eq V. V C L ( ) Cninue ineging emlely. b b Tube Vlge Ue he me equin Eq (V.0). V b b ( ) n hi he he RS (iin ) mve uw, he in (iin ) ninue mving inw. Evenully inee unil i vlue ehe he eeing vlue. 4

15 We mke he umin h he RS i uiienly enue when i ehe he in, h i veeue i negligible. n h e, he in my n be uhe uw, bu will ninue mve inw, lhugh i inw ee my be gully eue. 4 Slw Cmein Phe n hi he he in ee i: ( ) 4 Q - (XX) Hee we hve inlue enegy lgin em in he equin min. The lm gin enegy m Jule heing; n le enegy hugh Bemhlung & line iin. Enegy gin em will en uh he in uw. Enegy l em will hve he ing ee. Uing Sie m eiiviy, he lm lumn: T eime he emeue, T, we ue: Q J R T 8 k whee ( DN 90Z R 3 T ) m whee Riin Tem The Bemhlung l em my be wien : 5

16 Q B.6x0 40 N ι ( ) 6 ρ 6 0 ; N x N i N m M Rembinin l em i wien : Qe x0 N i Z T T The line l em i wien : Q n L Q Q Q 3 ( ) 5 ( ) x0 N ZZ J i B n Q L T Q whee Q i he l we ginl he lm lumn Q By hi uling, i, exmle, he iin l e B Q L i evee, hi wul le lge vlue inw. n he exeme e, hi le iin lle, wih ging ily e, uh mll vlue h he lm beme que he uging iin, hu ing he iin l. Thi iin lle u iil uen.6 MA (he Pee-Bginki uen) eueium. F ge uh Nen Agn, beue inene line iin, he iil uen i eue even belw 00kA, eening n he lm emeue. Plm Sel Abin n niin m vlumei emiin ue emiin Plm el bin n vlumei (emiin eibe bve) ue emiin he inh lumn hve been imlemene in he llwing mnne. The hni exiin numbe (ee File 3 Aenix by N A D Khk) i wien llw: M.66 x 0-5 Z n 0.5 n i (Z T.5 ) wih T in ev, e in S uni The vlumei lm el-bin ein A i bine in he llwing mnne: A ( 0-4 n i Z) (T 3.5 )) A AB A A ( M) Tniin m vlumei ue emiin u when he bin ein ge m (n bin) wn e (e.78) when he emiin beme ue-like given by he exein: Q nxz n Z ( ) T whee he nn n i ken 4.6x0-6 nm wih numeil exeimenl bevin h hi vlue enble he mhe niin, in genel, in em we vlue m vlumei ue emiin. 6

17 Whee neey nhe ine jumen i me he niin in juing he nn h he ue emiin we beme he me vlue he bin ee vlumei emiin we he niin in. Beyn he niin in (wih A le hn e) iin emiin we i ken be he ue emiin we. Neun Yiel h: Ae m he llwing e (wih miiin eum) Pinh uen limiin ee in lm u S. Lee n S. H. Sw, Al. Phy. Le. 9, 0503 (008), DO: Cyigh (008) Amein niue Phyi. Thi ile my be wnle enl ue nly. Any he ue equie i emiin he uh n he Amein niue Phyi. Thi ile ee in (iin bve) n my be un h:link.i.glink?applab90503 Neun yiel i lule wih w mnen, hemnule em n bem-ge em. The hemnule em i ken : Y h 0.5n i ( 3.4) <σv>(ime inevl) Whee <σv> i he hemlie uin ein-veliy u ening he lm emeue, he ime inevl une niein. The yiel Y h i bine by umming u ve ll inevl uing he u inh. The bem-ge em i eive uing he llwing henmenlgil bem-ge neun geneing mehnim 7, ine in he een RADPFV5.3. A bem euen in i ue by ie in in hin lye le he ne, wih lm iuin geneing he neey high vlge. The bem ine wih he h ene lm he u inh lumn ue he uin neun. n hi meling eh nibuing he yiel i eime inl quniy n he yiel i bine n exein wih inliy nn. The yiel i hen libe gin knwn exeimenl in. The bem-ge yiel i wien in he m: Y b- ~n b n i ( ) (σ v b ) whee n b i he numbe bem in e uni lm vlume, n i i he in eniy, i he iu he lm inh wih lengh, σ he -ein he D-D uin ein, n- bnh 8, v b he bem in ee n i he bem-ge inein ime ume inl he ninemen ime he lm lumn. Tl bem enegy i eime 7 inl L inh, meue he inh inune enegy, L being he u inh inune. Thu he numbe bem in i N b ~L inh v b n n b i N b ivie by he u inh vlume. Ne h L ~(b ), h 4 ~ ~, n h v b ~U whee U i he iuin-ue ie vlge 7. Hee b i he he iu. We l ume enbly h U i inl V mx, he mximum vlge inue by he uen hee lling illy w he xi. Hene we eive: Y b- C n n i inh ((b ))σv mx () whee inh i he uen lwing hugh he inh he lw mein he; n e he inh imenin en h he. Hee C n i nn whih in ie we will libe wih n exeimenl in. The D-D -ein i highly eniive he bem enegy i i neey ue he ie nge bem enegy mue σ. The e mue V mx he e 0-50 kv. Hweve i i knwn 7, m exeimen h he in enegy enible he bem-ge neun i in he nge 50-50keV 7, n mlle lwe-vlge mhine he elevn enegy 9 7

18 ul be lwe 30-60keV. Thu lign wih exeimenl bevin he D-D ein σ i enbly bine by uing bem enegy equl 3 ime V mx. A l exeimenlly meue neun yiel Y n v inh w me mbining ll vilble exeimenl,4,,3,7,9-. Thi gve i Y n 9x0 0 inh 3.8 inh in he nge 0.-MA. Fm hi l libin in w hen 0.5MA, Y n 7x0 9 neun. The mel e 3 RADPFV5.3 w hu libe mue Y b- whih in u mel i he me Y n. Clumn egin Whee in he il RS he we hve e en egin ee mel, we nw llw he egin be iven ully by he lm eue. ( ) - (XX) ρ 4 Ciui uen equin V C b L b () ( R ) - (XX) Equin (XX), (XX) n (XX) e inege ule equin, n. A eh e Q he vlue i l evlue bve. The l enegy ie by Bemhlung (Q B ) B n line iin (QL) my l be evlue. Vlge u eminl V b b (C) R nbiliy eineimene n inlue in lw mein he Fm exeimen, i i well knwn h e bie ei (ew n), he quieen lumn i ily bken u by inbiliie. One ee i huge ike vlge, illy beve u ube eminl. Thi vlge ike i enible iving in bem (w iein) n REB (negive iein, u he ne) wih enegie yilly 00keV. We ul mel hi by inluing uible ime vying eineimene in he equin; n juing hi unin ui he beve vlgebem enegy heii. Thee i mliin hi nnmlu eine in u melling. we inlue hi eine l 8

19 Q in he jule heing em in he in min Eq (XX), he uen inee in J will blw he in uw, leing huge negive vlge ike ; n exeiene exeimenlly. The mel my be me elii i he mmen inuing he nnmlu eine, he in min i en, even llwe ninue inw, he mgnei iel in uh mll Mgnei Reynl Numbe iuin will iue inw n in blw-u! The inl eul hi inbiliy mehnim i he beking u he u inh in lge exne uen lumn. 5 Exne Clumn Axil Phe We mel he exne lumn xil he (3,4) in he llwing mnne. n he exne lumn he we ume h he uen lw unimly m ne he in unim lumn hving he me iu he ne n lengh. The nmlie equin (me nmliin in xil he): ι βι e δι ι Ciui uen: β β ( ) e whee e Min: whee h e α ι e h h ( ) [ ( ) ] ( ) 4 The iniil niin ι n ι e he l vlue ι n ι m he l he. The iniil vlue i whee i he l lengh he u lumn, bu nmlie, he hn. Reeene J W Mhe. Phy Flui, 8, 366 (965) S Lee, T Y Tu, S P M, M A Ei, A V Ghl, K H Kwek, S Mulyn, A J Smih, Suyi, W U, M Zkullh. Ame J Phy 56, 6 (988) 3 S Lee. Exeimen wih he CTP-UM 3.3 kj Plm Fuin Filiie. CTP-H4-SMR5544 (99) 4 S Lee. EEE Tn n Plm S, 9, 9 (99) 5 M H Liu, S Lee. SXR Riin Melling Nen Plm Fu. nen Cnge n Plm Phy, Pgue, June 998 9

20 6 S Lee, P Lee, G Zhng, X Feng, V Gibkv, M Liu, A Sebn, T K S Wng. EEE Tn Plm S, 6, 9 (998) 7 S. Lee & S. H. Sw, Alie Phy Lee 9, 0503 (008) 8. S Lee & S. H. Sw. J Fuin Enegy. DO: , ublihe nline: 0 Febuy S. Lee, S. H. Sw, P. C. K. Lee, R. S. Rw n H. Shmi. Cmuing Plm Fu Pinh Cuen m Tl Cuen Meuemen; Alie Phy Lee, in uin Vl 9, ue, Mh