Alpha Particle Dynamics in Muon-Boosted Fusion Propulsion System

Transcription

1 44th I/SM/S/S Joit Propulsio Cofr & xhibit - 3 July 8, Hartford, CT I lpha Partil Dyais i Muo-Boostd Fusio Propulsio Syst Trry Kaash, Riky Tag ad l D. Gallior 3 Uivrsity of Mihiga, rbor, MI, 489 I a prvious papr, w dostratd that gativ uos rsultig fro atiproto aihilatio i a rlativly old dutriu-tritiu (DT) plasa ofid i a gasdyai irror (GDM) a rsult i atalyzig o avrag ovr fusio ratios. Th alpha partils produd by ths ratios ould otribut sigifiatly to hatig th bakgroud plasa toward igitio. I fat, it was poitd out that o th basis of rgtis oly, uo-atalyzd fusio would rdu th aout of atiprotos rquird to ahiv throular bur by about 6%. This sario, howvr, dos ot addrss th issu of alpha partil ofit i th GDM, ad thrby lavs op th qustio of thir tru fftivss i providig th hatig otd abov. I this papr, w addrss this probl by otig that, as thy slow dow, ths alpha partils a sap fro th syst. W ddu xpliit xprssios for alpha partil dsity as a futio of rgy, ad alulat th a rgy of ths partils allowig siultaously for slowig dow ad sap as rfltd by th ofit ti. ssuig that th alpha partils slow dow priarily o th ltros, as is th as i rlativly old plasas, w fid that uo atalyzd fusio is idd fftiv i hatig th plasa i a GDM dvi. th olatur ubr of atalyzd fusio pr gativ uo alpha partil rgy iitial (birth) rgy of alpha partil thral rgy a alpha partil rgy ltary harg L plasa lgth l Λ Coulob logarith partil ass a ofid alpha partil dsity ubr dsity ( ) alpha partil rgy distributio futio R plasa irror ratio T ltro tpratur τ ti ostat, or ofit ti τ uo lifti v Z oorgti partil vloity partil harg stat Stph S. ttwood Profssor ritus of ular girig ad Radiologial Sis, 355 Boistl Blvd., rbor, MI 489, I ssoiat Fllow. Graduat Studt, Dpt. of rospa girig, 3 Bal v., rbor, MI 489, I Mbr. 3 rthur F. Thurau Profssor of rospa girig, 3 Bal v., rbor, MI 489, I ssoiat Fllow. ria Istitut of roautis ad stroautis 947 Copyright 8 by th ria Istitut of roautis ad stroautis, I. ll rights rsrvd.

2 I. Itrodutory Rarks hav show i prvious work that i atiproto-driv fusio propulsio systs, plasa hatig rsults W fro fissio fragts as wll as fro th aihilatio produts produd by atiproto aihilatio i U 38 targts. W hav also idiatd that so additioal sigifiat hatig os about as a rsult of uo atalysis i a dutriu-tritiu (DT) plasa whrby a gativ uo a uiquly attah itslf to both D ad T ios, thrby allowig th to udrgo fusio ratios ad rlasig rgti alpha partils ito th plasa. Suh atalysis rsults i or tha fusio ratios durig th lifti of th uo, ad th rsultig alpha partils provid sizabl aout of hatig towards igitio. Th assuptio ad i thos studis is that th alpha partils i th ofit dvi th gasdyai irror (GDM) dposit thir rgy through ollisios ad ot sap whil doig so. I this papr, w addrss this qustio by allowig for sap whil ths partils slow dow. W assu that alpha partil ofit follows that of th ios, ad ddu th appropriat xprssios for thir vloity distributio ad a rgy, as wll as thir ofit ti i th GDM. W fid that dspit partil losss, th fratio of alpha partils that ar ofid a still otribut a sigifiat aout of plasa hatig as suggstd arlir. What follows is a athatial ofiratio of ths prditios. II. rgy Distributio Th ubr of alpha partils i a itrval of rgy Δ is uit rgy. If a loss hais has a ti ostat τ ( ) d dt [ ( ) Δ] ( ) ( + Δ) s ( ) Δ, whr ( ), th i stady stat w hav is th ubr dsity pr ( ) Δ ( ) d d () dt dt τ + Δ Rarragig ad usig th dfiitio of a drivativ, w obtai th followig govrig diffrtial quatio for th rgy distributio of alpha partils. ( ) ( ) ( ) ( ) d d d + + ( ) + () dt τ dt dt τ Itgratig q. () ovr th rag of rgis to, whr is th iitial rgy (i.. birth rgy) of th alpha partils, w obtai th followig. Th first tr of th itgrad a b rwritt as follows d ( ) ( ) dt + xp d (3) d d ( ) dt τ dt d dt d dt d d d l (4) dt ad a b radily itgratd yildig th followig xprssio for th alpha partil rgy distributio. ( d ) dt ( ) ( ) xp d (5) d ( ) d τ dt dt ow osidr alpha partils produd via uo-atalyzd fusio, th iitial rgy distributio would b giv by th followig. ria Istitut of roautis ad stroautis 947

3 ( ) ( ) τ ( d ) dt (6) whr is th gativ uo ubr dsity, τ th uo lifti, ad is th ubr of atalyzd fusio (i.. ubr of alpha partils bor) pr gativ uo. d dt rprsts th rat of dras of alpha partil rgy ad a b xprssd by th followig, with th first tr dotig rgy loss to th plasa ltros ad th sod tr loss to th ios du to Coulob ollisios. d + (7) dt ad ar offiits that dpd o th iidt ad targt partils, as wll as plasa dsity ad tpratur. - ( l Λ) C [ ] 8 π (8a) 4 Z Z s 3 a 3 T 9 (.6 ) 3 3 kv rg C a (8b) whr ad ar rsptivly th ass of th iidt partil (i.. alpha partil) ad th ltro. Siilarly, Z ad Z ar th harg stat of th iidt partil (i.. alpha partil) ad th ltro, rsptivly. is th ltro dsity, ad T is th ltro tpratur. quatio (8a) is writt i th CGS syst, ad all th quatitis hav th stadard CGS uits, with th xptio of th ltro tpratur T. For ovi, T i q. (8a) has uit kv. Th ovrsio fator C a aks xpliit th ovrsio to th CGS syst. Fially, l Λ is th Coulob Logarith giv by th followig for a DT plasa, -3 [ ] l Λ 4 log (9) T Siilarly for, q. (a) is writt i th CGS syst, ad all quatitis hav thir stadard CGS uits. Th ovrsio fator C b surs that has th orrt rgy uit of kv i ordr to b osistt with th othr quatios. 4πZ Z ( l Λ) [ V] 3 ria Istitut of roautis ad stroautis 947 C 4 i i b i 9 (.6 ) kv s 3 3 (a) 3 kv rg C b (b) Substitutig qs. (6) ad (7) ito q. (5) yilds th followig rgy distributio for alpha partils produd via uo-atalyzd fusio, whr has uit kv. ( ) ( ) τ + d xp () 3 ( ) ( ) + τ

4 To valuat q. (), w d th ti ostat (i.. ofit ti) τ ( ) GDM, igorig th abipolar pottial, is as follows, s. Th ofit ti for th RL τ ( ) () v Hr R is th plasa irror ratio, whih is th ratio of th agti fild s by th plasa at th irror to that at th tr. Th oorgti partil vloity is giv by q. (3) C v (3a) 9 C.6 rg kv (3b) whr is th ass of th alpha partil, ad C is a uit ovrsio fator allowig i q. (3a) to b xprssd i kv i ordr to b osistt with q. (). ll th othr quatitis i qs. () ad (3a) hav th stadard CGS uits. Th hoi of th CGS syst hr is arbitrary; SI uits ay b usd istad i qs. () ad (3a), i whih as C would b odifid to rlat Joul to kv. Substitutig ths quatios ito q. () yilds th fial xprssio for th rgy distributio for alpha partils produd via uo-atalyzd fusio isid th GDM. ( ) ( ) τ C xp d 3 RL [ kv ] III. ltro Hatig Oly Th aalytial solutio to th full itgral i q. (4) is vry opliatd. For a rlativly old plasa,.g. at th ioizatio tpratur, a b svral ordrs of agitud largr tha, ad thrfor, ltro hatig doiats, whih is what w xpt for a old plasa. If w visio a GDM syst whri ths alpha partils produd via uo-atalyzd DT fusio ratios otribut to th iitial phas of th plasa hatig, it is rasoabl to assu th bulk of thir rgy big dpositd ito th plasa ltros. W a thrfor siplify th itgral i q. (4) aordigly by assuig. (4) ( ) ( ) τ C d xp RL (5) Th rsultig itgral a b asily valuatd, yildig th followig distributio futio. C ( ) xp ( ) τ xp τ RL { ( )} (6) whr w hav dfid th followig quatity. C RL > - [ kv ] (7) 4 ria Istitut of roautis ad stroautis 947

5 Isptig q. (6), w s that th distributio bhavs as follows. ( ) ~ (8) For rlativly sall rgy, doiats, whras for larg, th xpotial doiats. Th iiu of th distributio ours at a rgy i. 4 i (9) Si th rgy of th alpha partils will b boudd by th (thral rgy) ad (iitial rgy at birth, i MV), oly this portio of ( ) is aigful. Usig typial ordrs of agitud for th dfid quatity, i.. -3 to -5 for a ds old plasa, w s that our distributio lis sigifiatly to th lft of th iiu, whr doiats. Figur shows a rprstativ plot of th distributio futio for this rag of rgis. Figur. Typial profil for th alpha partil rgy distributio futio.. Cofid lpha Partil Dsity To obtai th total dsity, w itgrat th distributio, q. (6), ovr all rgis btw th lowr ad uppr bouds. th ( ) d τ hag of variabl trasfors th itgral i q. () ito th d () 5 ria Istitut of roautis ad stroautis 947

6 τ τ th z z z dz z dz + th z z dz () ah of th itgrals i q. () is dfid as th xpotial itgral futio ad is dotd by i. Thrfor, th total dsity assuig ltro hatig oly is giv by th followig. τ -3 [ i( ) i( th )] [ ] () B. Ma Kiti rgy Th a alpha rgy a b alulatd as follows. th ( ) d τ th d I (3) Th itgral I a b radily valuatd by first akig a variabl substitutio by parts. I τ d th fial xprssio for th a alpha partil rgy is th [ ( ) ( th ) ] th ( ) ( th ) [ i( ) i( th )] x ad th usig itgratio [ kv] (5) C. Sapl Calulatios W osidr a dutriu-tritiu (DT) plasa with dsity at a iitial tpratur of 3.6 V, orrspodig to th ioizatio pottial of th propllat. Th syst is atiproto driv. First, at rst aihilatio of atiprotos i uraiu-38 targts auss fissio at arly % ffiiy.,3 Th rsultig fissio fragts ad aihilatio produts, aly pios ad thir day produt uos, otribut to th hatig of th plasa. I additio, i a DT plasa (v if it is old), ah gativ uo a o avrag atalyz approxiatly DT fusio ratios, ah rlasig a alpha partil of 3.5 MV of kiti rgy that furthr otributs to th iitial phas of plasa hatig. For a GDM with plasa irror ratio of 5, a plasa lgth of trs (ot: this alulatio is ot ssitiv to ths two quatitis), ad a atiproto dsity of.7-3 (basd upo hatig rquirts ot addrssd i this papr ad orrspodig osrvativly to th sa dsity of gativ uos, whih i turs yilds a iitial alpha partil dsity of ), th ubr of alpha partils big ofid is about Th a rgy of ths ofid alpha partils is roughly 94 kv. To dtri th hag i plasa tpratur, w osidr a sipl rgy bala. (4) 3 ( T T ) ( ) i (6) 6 ria Istitut of roautis ad stroautis 947

7 whr i is th iidt partil dsity (i.. alpha partils i th urrt aalysis), ad th subsript dots iitial valus. If w assu th ofid alpha partils dposit alost all of its rgy ito th plasa ltros, i.. slow dow o th ltros fro thir birth rgy of 3.5 MV to a fial avrag kiti rgy ( 3 ) T.4 V, orrspodig to a tpratur of 3.6 V, th hag i th ltro tpratur is ΔT 784 V. This rprsts th axiu hatig produd by ths alpha partils. ltratly, if w ak a or osrvativ stiat ad assu th ofid alpha partils slow dow o th ltros util thy rah thir a kiti rgy of 94 kv, th th orrspodig hag i th ltro tpratur is ΔT 78 V. Fro q. (6), w s that i Δ T ~ Δ (7) Si Δ is or or lss fixd, th iportat fator is th dsity ratio. Irasig this ratio ithr by irasig atiproto dsity or drasig ltro dsity or both a rsult i a Δ T of ulti-kv s, for ista. Of ours, du to th hatig rquirts, ths two dsitis ar ot ssarily idpdt. For xapl, hagig th ltro dsity will hag th iiu atiproto dsity rquird, aig th gativ uo dsity, ad h th ofid alpha partil dsity, will hag as wll. Howvr, o a iras th atiproto dsity byod th iiu rquird valu ditatd by th hatig rquirts to produ a largr Δ T, if this is dsird ad th assoiatd iras i ost of obtaiig ad ofiig th atiprotos is ot prohibitiv. IV. Colusio I this papr, w hav drivd a rgy distributio (or quivaltly a vloity distributio) for th alpha partils produd via uo-atalyzd fusio isid th GDM. Th distributio iorporats a ti ostat to addrss partil losss du to sap fro th syst durig slowig dow. W hav drivd fro th distributio th ubr dsity for th ofid alpha partils, as wll as thir a rgy, by assuig that th ajority of rgy trasfr is to th plasa ltros, a valid assuptio for a rlativly old plasa. W foud that although thr ar partil losss, th ubr of alpha partils raiig ad th hatig thy otribut ar vrthlss sigifiat. For a giv plasa dsity, w a iras th atiproto dsity to iras th aout of hatig as a rsult of a iras i th ofid alpha partil dsity. For ista, i th abov alulatios, doublig th aout of atiprotos will rsult i ~.5 kv iras i th ltro tpratur. Th a rgy ad th prtag of alpha partils ofid, howvr, will rai th sa for a giv plasa dsity. othr way to iras th otributd hatig is to iras th alpha partils utilizatio. Brif alulatios showd that utilizatio irass as th plasa dsity drass. For ista, wh 6-3, th prtag of alpha partils ofid log ough to hat th plasa irass 4-fold opard to th alulatios i th prvious stio, ad th assoiatd hatig irass sigifiatly as wll. Th tradoff, howvr, is that th plasa dyais isid th GDM ditats a rathr rapid iras i th plasa lgth with drasig plasa dsity, ad th syst soo bos prohibitivly assiv. kowldgts This work was supportd by a S GSRP fllowship through Marshall Spa Flight Ctr. Rfrs Kaash, T., ad Tag, R., Muo-Boostd Fusio Propulsio Syst, 43 rd Joit Propulsio Cofr, Ciiati, OH, 7. I Hofa, P., t al., Fissio of Havy uli Idud by Stoppd tiprotos. I. Ilusiv Charatristis of Fissio Fragts, Physial Rviw C, Vol. 49, 994, pp Ki, Y. S., t al., Fissio of Havy uli Idud by Stoppd tiprotos. II. Corrlatios btw Fissio Fragts, Physial Rviw C, Vol. 54, 996, pp ria Istitut of roautis ad stroautis 947