Modelling of Fission Chambers in Current Mode Analytical Approach

Transcription

1 Modlling of Fission Chmbs in Cunt Mod Anlyticl Appoch Sébstin Chbod,, Gbil Fioni, Alin Ltounu, Fédéic Mi DSM/DAPNIA/SPhN, CA-Scly, 99 Gif-Su-Yvtt, Fnc DSM, CA-Scly, 99 Gif-Su-Yvtt, Fnc Abstct A comphnsiv thoticl modl is poposd to xplin th functioning of fission chmbs optd in cunt mod, vn in vy high nuton fluxs. Th clibtion cuvs clcultd s function of bsic physicl pmts s fission t, gs pssu nd gomty of th chmbs. Th output cunt t stution is pcisly clcultd, s wll s th mximum voltg to b pplid in od to void vlnch phnomn. Th lctic fild distotion du to th spc chg phnomn is lso stimtd. Within this modl, th chctistic sponss of fission chmbs coctly poducd, in gmnt with th xpinc fdbck obtind t th ILL/Gnobl High-Flux cto. Ky wods: fission ioniztion chmb, cunt mod. PACS: 9.40.Cs, 8.4.c. Intoduction Fission ioniztion chmbs widly usd s nuton monitos in iditing nvionmnts such s nucl ctos, ccltos nd mdicl fcilitis. Thy cn b usd in puls mod, wh ch lctonic puls inducd by nucl fission is countd vnt by vnt. Nvthlss, in high nuton fluxs bov 0 4 n.cm -.s -, th puls pil-up inducd by th high fission t quis cunt mod cquisition, wh ch singl vnt is not nymo individulisd. In od to cy out on-lin msumnts of tnsmuttion ts of ctinids t th Mini-Inc nd Mgpi instlltions [-6], w hv cntly dvlopd b sub-minitu cylindicl fission chmbs dsignd to stnd high tmptus nd nuton fluxs up to svl 0 5 n.cm -.s -. Bsid thi us fo nucl wst tnsmuttion studis, ths dtctos usful tools fo in-co nuton flux dignostics of Gntion-IV nucl systms. Aft svl xpimnts in high nuton fluxs of bout 0 5 n.cm -.s -, w hv obsvd tht th sponss of th chmbs s function of th pplid voltg, nmly th clibtion cuvs, ptubd. Th shps of th clibtion cuvs diff significntly fom thos obtind duing idition in nuton fluxs on od of mgnitud low. W hv xpimntlly noticd tht th pssu of th filling gs nd th gomtis of lctods hv stong influnc on th sponss of th dtctos, with diffncs tht ccntutd in high-intnsity nuton fluxs. In od to hv cl undstnding of th obsvd phnomn, w hv dvlopd n nlyticl modl to study nd pdict th volution of th clibtion cuvs s function of diffnt physicl pmts, s gs pssu, gs composition, pplid voltg, duing vious conditions of idition, fom low to high intnsity nuton fluxs. -mil: schbod@c.f, Phon: , Fx: b in collbotion with CA/DN/SPX Lbotoy nd PHOTONIS Compny

2 This ppoch is dscibd in th psnt pp. In th fist pt, th bsic qutions usd to modl th functioning of fission chmbs in cunt mod dtild. In scond pt, this thoticl fmwok is pplid to th clcultion of th clibtion cuvs.. Modlling of fission chmbs Cylindicl fission chmbs md of two coxil lctods nod nd cthod sptd by filling gs, s shown in figu 5. Th nod is usully cotd with fissil lmnt. Und idition, nutons induc fission ctions insid th dposit nd high ngy fission poducts bout 90 MV/ch on vg fo 35 U mittd in opposit dictions. Thus, on is bsobd in th nod whil th scond cosss th int-lctod spc, ionizing th filling gs on its pth nd consquntly gnting high numb of lcton-ion pis. Whn voltg is pplid, n lctic fild is gntd btwn th two lctods, involving migtion of chgs. Th collctd chgs sponsibl fo th ction of n lctic cunt. Th lyout of this cunt ccoding to th voltg pplid givs chctistic cuv, know s th clibtion cuv. In th following, w will consid stndd cylindicl fission chmb with 98.5 % pu 35 U dposit s CFUT-C3 c chmbs usd in th fmwok of th Mini-Inc nd Mgpi pojcts, s figu 5... Clcultion of th chg pi dnsity ctd by th fission poducts In this sction, w will vlut th dnsity N of lcton-ion pis ctd p unit of tim by th fission poducts in th int-lctod spc. To simplify ou clcultions, w not N fst, th numb of fission poducts lsd p scond nd p unit of t th nod. In cylindicl gomty, on cn obtin N fst τ f,. π h wh h is th lngth of th dposit which ptilly covs th nod s figu fo th nottions. is th nod dius. τ f is th fission t, i.. th numb of fissions tht tk plc insid th dposit p unit of tim. Lt us lso not Xd, th vg numb of pis ctd by fission poduct p unit of lngth tvlld in th gs. X dpnds on th distnc d covd by th fission poduct in th int-lctod spc. A fist ppoximtion consists in supposing tht ll th fission poducts lv th nod with puly dil spd v. In cylindicl coodints, v cn b wittn v u + v θ u θ + v z u z. Consquntly, this ssumption fixs to zo th componnts v θ nd v z. Within this simplifid fmwok, on cn obtin n xpssion fo N tht dpnds only on th coodint nd tht cn b wittn with. N fst X ' N,. c CFUT-C3 chmbs mnufctud by th PHOTONIS Compny.

3 A scond ppoch, dvlopd by Poujd t Lbun [7], consists in ssuming tht ll fission poducts lv th dposit ppndicully to th nod xis. Th vlocity componnt v z is thus fixd to zo nd th uthos hv shown tht N dpnds onc mo only on nd cn b wittn ccos ccos N fst cos θ N X ' dθ,.3 π + cos θ with + cos. ' θ Fig.. Schm of n nod nd its dposit. If th distnc d covd by fission poduct is smll s compd to its mn f pth, Xd mins bout constnt ll long th tjctoy. Thus w hv Xd X 0. Within this fmwok, th.3 intgl cn b clcultd nd w hv obtind th following xpssion N N π fst X 0 ctn..4 W cn vify tht N fst X 0 N + π..5 N N X fst 0 Th fist two ppochs sttd fom ppoximtions nd ld to simpl xpssions fo N. In od to clcult it pcisly, lt N ft b th numb of fission poducts jctd p scond by n infinitsiml sufc lmnt ds of th nod. N ft cn b wittn in cylindicl coodints N ft N fst dz' d ',.6 θ 3

4 wh,θ',z' th cylindicl coodints of ds. Th numb of lcton-ion pis ctd p scond, p sufc lmnt ds, in th infinitsiml volum dτ suounding th point P,θ,z in th gs is givn by d d N N X ' d' Ω ft,.7 π wh dω is th solid ngl cosponding to th volum dτ sn fom th lmnt ds. ' is th distnc btwn th point P nd th lmnt ds. In sphicl coodints, w cn wit nd d N thus bcoms d dτ ' sin ϕ d' dϕdψ,.8 dω sin ϕ dϕdψ N ft X ' sin ϕ d' dϕ ψ N..9 π d W obtin th dnsity of pis dn ctd ound th point P by th lmnt ds by dividing d N by dτ dn d N N ft X ' N fst X ' dz' dθ '..0 dτ π' π ' In cylindicl coodints, ' θ θ. nd thus + cos ' + z z' N fst X ' dz' dθ ' dn, θ, z.. π + cos θ ' θ + z z' To obtin N,z, th dnsity of pis ctd by th fission poducts p unit of tim, w hv only to intgt. xpssion ov th sufc of th nod, N, z ccos / + θ θ ' ccos / + θ z' h / ccos / h / h / θ ' ccos / z' h / N N fst fst X ' π π X ' + cos θ ' θ + z z' + cos θ ' + z z' dz' dθ ' dz' dθ '..3 If w ssum X X 0, w cn gt out th pmt X fom th intgl nd wit finlly N fst X 0 N, z F, z,.4 π 4

5 with F, z ccos / h / + cos θ ' + z z' θ ' ccos / z' h / dz' dθ '..5 W hv dwn on figus nd 3 th function F clcultd fo h cm nd.5 mm. Fig.. Compison btwn th xct fomul of F, th / ppoximtion nd th Poujd xpssion.5 mm, h cm. Fig. 3. volution of F,z with insid th dposit zon fo vious z.5 mm, h cm. 5

6 Fig. 4. volution of F,z with z fo vious.5 mm, h cm. W mk on figu tht th ppoximtd xpssions ld to consqunt undvlution of th numb of ctd pis. On figus 3 nd 4, w notic tht F,z mins clos to F,0 in th dposit zon nd flls butlly byond. Consquntly, w cn modify th F xpssion to limint th z componnt F, z zmx F F, z dz insid th dposit zon i.. fo - h/ z h/ h zmin,.6 0 outsid th dposit zon i.. fo z -h/ nd z h/ wh z min nd z mx th limits in z of th int-lctod spc. Thn, w ssum tht N, z N fst X 0 N F insid th dposit zon π..7 0 outsid th dposit zon Fig. 5. Schm of CFUT-C3 clss chmb. Fo CFUT-C3 chmb, z min nd z mx spctivly qul to cm nd 0.7 cm. W thus obtin th following volutions of <F>,z nd F, dwn on figus 6, 7 nd 8. 6

7 Fig. 6. volution of F,z nd <F>,z with z.5 mm, h cm. Fig. 7. volution of F,0 nd F with.5 mm, h cm. 7

8 Fig. 8. volution of F with fo vious diuss h cm. W vify using.6 xpssion tht zmx zmin zmx, z dz F, z dz hf F..8 zmin Thus, s it will b shown in sction 3.., th o inducd by.7 ppoximtion on th cunt t stution is null... Cofficint X 0 Th X 0 cofficint s sction. psnts th numb of lcton-ion pis ctd by fission poduct p unit of lngth tvlld in th int-lctod spc. Using SIM softw [8], w hv clcultd fo Agon: X 0 pis.m P[b],.9 whn th chmb is filld t mbint tmptu T 300 K. P is th gs pssu givn in b. This sult is clos to th vlu commndd by th ICU pot [9] which is woth.0 8 P [b] pis.m -. Th SIM clcultions hv bn conductd by vging th loss ngy in th gs ov th complt distibution of 35 U fission poducts. W hv lso ssumd tht th vg ngy W ncssy fo fission poduct to ct pi is 6.4 V. In xpimnts, on cn indd find [9] 6.4 V fo 5.3 MV α W 6.66 V fo.8 MV potons ± 0.5 V fo lctons on vg 8

9 Lt us not now i th fist ioniztion ngy of th gs 5.75 V fo Agon, η th numb of lctons poducd nd η x th numb of xcitd Agon toms notd A * ctd by th fission poducts. On cn obtin fo th gss [9] W η x,7,8 i.. 0,4 0,5 η W cn notic tht th is on vg on xcitd tom fo two lctons poducd in Agon. In ddition, Bhl t l [0] giv svl thshold ngis fo som A + - ctions tbl. ctions Thshold ngy [V] xcittion 3,5 A ,75 A * + - 9, A ,98 Tb.. Thshold ngis fo som A + - ctions. To ct A + ions, w mk tht high ngis quid. As consqunc, thi numb will b much low thn A + ions nd w cn nglct thm. Finlly, th ICU pot [9] psnts th volution of th W cofficint of som gss ft ddition of svl pollutnts. In pticul, th ddition of Kypton, wt o i in Agon involvs diminution of its W cofficint. Fo Hlium t pssu of on b, th ddition of of Xnon, Agon, Kypton o CO involvs n incs in X 0 of fcto.4 to.5. A concnttion of % of Agon in th sm Hlium inducs 4 V duction in its W cofficint, whil % Agon pollution is ccompnid by 5 V fll of W. This nonngligibl incs of th numb of pis ctd in psnc of quntitis, vn smll, of pollutnts is phnomnon known s th Jss ffct.. 3. Bsic chg tnspot qutions Th tnspot of lctic chgs insid th chmb obys som bsic qutions. Lt us not ρ nd ρ, th lctic dnsitis spctivly ssocitd with lctons nd Agon ions. Lt us intoduc lso <v > nd <v >, th vg dift vlocitis of lctons nd ions. Th chg consvtion qutions ld to th following systm n t n t + div + div ρ v ρ v t t SOUC SOUC + t + t LOSS LOSS.. Th t SOUC nd t LOSS tms th ction nd disppnc tms. Thy psnt th dnsitis of lctic chgs which ctd nd dstoyd p unit of tim in gs. Thi xpssions will b dtild in th following sctions. In cunt mod, th chmbs wok in sttiony stt. Th tmpol divtivs of. systm thus vnish nd it mins 9

10 div div ρ v ρ v t t SOUC SOUC + t + t LOSS LOSS..3 In cylindicl fission chmb, nod nd cthod two coxil cylinds s figu 5 of spctiv diuss nd. W pply voltg V btwn nod nd cthod. This diffnc of lctic potntil gnts n lctic fild tht obys Mxwll-Guss qution nd is wittn ρ + ρ div ε dl V In cylindicl gomty, if w suppos tht th dnsitis ρ nd ρ dpnd only on th dil coodint, notd s sction., on cn thn wit u ρ + ρ..5 ε 0 d V Th lctic fild is dictd towds cthod. Consquntly, ions migt towds cthod whil lctons mov towds nod. As consqunc, w hv v v v + v u u..6 nd w obtin th following systm ρv t ρ v t SOUC SOUC t t LOSS LOSS,.7 wh v nd v spctivly th vg dil pojctions of th spd of lctons nd Agon ions. Duing this study, w will only consid A + ions s sction.. Th.7 systm is thus ducd to qutions instd of on qution fo lctons nd on p spcis of A n+ ions. Moov, th bsolut vlus of th ction nd loss tms fo lctons nd A + ions bcom qul, th disppnc of ngtiv chg on lcton involving th disppnc of positiv chg on A + ion. Now, if w not 0

11 ρ n ρ n t SOUC t t LOSS t LOSS SOUC T T L S,.8 th.7 systm cn b wittn nv TS T n v TS TL L..9 This systm llows us to clcult th dnsitis n nd n of lctons nd ions in gs. Th T S nd T L tms cospond h to th dnsitis of lcton-ion pis spctivly ctd nd dstoyd p unit of tim in int-lctod spc. Th cunt I dlivd by th chmb cn b wittn n v + I n v. ds..30 S W cn chck tht th cunt I is constnt of th.9 systm. In cylindicl gomty, th.30 xpssion cn b wittn in th following wy I π h n v + n v..3 Th lctods connctd to n lctic supply ntwok. Thus, w hv in sttiony stt nd xpssion.3 lds to n n 0,.3 0 I π h n v πh n v..33 To conclud this pt, w simply not tht th coupld solution of th.5 nd.9 systms giv th functions n, n nd ccoding to th voltg pplid to th lctods. Thus, using.3 o.33 xpssions, w now bl to clcult th cunt I s function of th voltg V, i.. to dw th nti clibtion cuv.. 4. Ction nd loss tms Th T S nd T L tms pping in systm.9 spctivly clld th souc nd loss tms. Thy psnt th dnsitis of chg pis ctd o dstoyd p unit of tim insid th chmb Souc tms

12 Th T S tm is minly th sum of two contibutions. Fist, tm T S du to th ioniztion of th gs by th fission poducts. T S is qul to N nd hs bn clcultd in dtil in sction.. Scondly, tm T S sulting fom scondy ioniztions du to lctons which hv cquid nough kintic ngy in th lctic fild. Th T S tm is usully wittn T αn v,.34 S wh α is th Townsnd fist ioniztion cofficint. Whn th vitions of th lctic fild wk ov n lcton mn f pth, α is only ltd to nd th mny thoticl o smi-mpiicl fomuls which llow its clcultion on limitd intvls. W will tin in pticul th Townsnd fomul α P A BP,.35 wh P is th pssu of th filling gs. This fomul is in th good gmnt with xpimntl sults nd hs physicl justifiction. Th cofficints A nd B cn indd b clcultd ccoding to pmts lik th mn f pth l 0 of n lcton in gs o th vg fction of ngy lost by n lcton duing collision with n tom o molcul of gs. Zstwny [] cid out compiltion of th vlus tkn by A nd B fo mny pu nd mixd gss. Most of th smi-mpiicl xpssions of α poposd fo low gs pssus. W hv consquntly usd th BOLSIG softw [] to obtin stimts t tmosphic pssus s figu 9. Fig. 9. tios of th ioniztion cofficint α ov pssu P, clcultd using BOLSIG softw fo vious pu nd mixd gss. Lkshminsimh t l [3] giv lso xpimntl vlus of α on lg ng of fo svl gss, in pticul Agon Loss tms

13 Th T L tm is minly du to th combintion of lctons with ions. Ths pocsss of cptu pocd in th stps, clssifid by siz scl initil: th jctd lcton combins with th ion fom which it coms fom. Onsg [4] poposd fomul giving th fction of pis scping fom this initil combintion ccoding to th lctic fild. This mod of combintion plys only t high pssus 0 to 00 tmosphs. column: th pis cn thn combin insid th ionizd tcks lft by th fission poducts in gs. Ths tcks hv indd high dnsitis of chgs. voluminl: th most gnl pocss. Th lctons migt in gs und th ffct of th lctic fild nd cn combin with ny ion thy mt. Fo pssus clos to th tmosph, th dominting pocss is th voluminl combintion. Th T L tm cn thus b wittn s follows T kn n,.36 L wh k is gnlly clld th combintion cofficint. Th stimtion of th cofficint k is difficult nd it xists only vy fw dt. Lngvin 903 givs thoticl xpssion of k k th 4π µ + µ 4πµ,.37 ε ε ε ε 0 0 s µ th mobility of lctons is much high thn µ th mobility of ions. This xpssion is vlid whn th mn f pth l 0 is smll in font of chctistic distnc 0 givn by 0 kt,.38 ε wh ε is qul to t pssu of 0. MP [5]. Shinsk t l [6] hv thft shown, fo Agon t tmptu T of 300 K nd pssu P clos to 00 b, tht k xp k th By ssuming tht th cofficint k vis in P / [7], w hv thn cos stimt of k t low pssus k P..40 k th Fo ducd lctic filds /P high thn V.m -.b -, µ is bout m.v -.s - t pssu of on b s sction.7.. W hv thus fist stimt of k t on b: k m 3.s -. Biondi [8] msus lso k m 3.s - t P 0.08 b nd T 300 K fo Agon. Fo Non, h obtind k m 3.s - t P 0.03 b nd T 300 K. By pplying th lw in P /, w obtin scond stimt of k t on b: k m 3 /s. This sult is clos to th 3

14 vlu xtpoltd fom th sults of Shinsk [6]. Fo Non, w obtin t on b: k. 0 - m 3.s Discussion on th shp of clibtion cuv In cylindicl gomty, th cunt I obys.3 qution. Th dnsitis n nd n givn by th systm.9 with th boundy conditions.3. Using th xpssions of th ction nd loss tms, w finlly wit nv N knn + αnv n + v N knn αnv This systm cn b intgtd nd w obtin..4 I N kn n αn v π h + d..4 qution.4 cn b dissocitd in th contibutions I I I I I 3 + I πh πh πhk + I N d α 3 n n d n v d..43 4

15 Fig. 0. combintion, stution nd vlnch mods in clibtion cuv. Th I contibution is constnt with th voltg V pplid nd cosponds to th intnsity dlivd in zon tht w commonly cll th stution pltu s figu 0. Now, if on ducs th voltg pplid t th lctods, th lctic fild will dcs nd th chgs will consquntly cicult mo nd mo slowly, ccumulting in th filling gs. Th dnsitis n nd n will thus incs nd th combintion tm I, which is ngtiv, will tk impotnc. As consqunc, th output cunt I will dop gdully blow th stution cunt I. This pticul mod is clld th combintion mod nd w cn indd obsv it t low voltgs, on th lft of th stution zon s figu 0. In th sm wy, whn th voltg incss, th lctic fild is gdully infocd. Th fist ioniztion cofficint αε, up to now qusi zo, will incs slowly, until ching thshold. 0 5 V.m - fo Agon t on b wh its gowth cclts butlly s figu 9. Consquntly, th I 3 contibution nd th totl cunt I will incs quickly. This fst incs is clld th vlnch mod nd fms on th ight th stution zon on clibtion cuv s figu 0. A clibtion cuv thus sults fom compomis btwn th mods, combintion, stution nd vlnch, whos ky pmts will b studid in dtil in this ticl.. 6. Spc chgs nd distotion of th lctic fild Th qutions which contol th lctic fild insid cylindicl chmb wittn ε 0 d V n n..44 5

16 6 Th fist qution cn b sily intgtd nd w obtin + d n n A ε W cn dmonstt thn tht ln ln ln 0 d n n V A ε..46 In bsnc of lctic chgs, w not tht ln V A,.47 nd w find th usul unptubd xpssion of th lctic fild u V ln..48 Th psnc of lctic chgs insid th chmb inducs distotion of th lctic fild, which dos not vy ny mo s /. W obsv in pticul, stting fom qution.46, tht n incs of th ion dnsity n involvs duction of th fild t nod. Th spc chgs phnomnon pps of cucil impotnc t high fission ts. W will hv th oppotunity to consid in dtil this ptubtion in th continution of this study.. 7. Dift vlocitis of lctons nd ions Th popgtion of th lctic chgs in th filling gs is govnd by th.9 systm. Tking into ccount th diffusion pocsss, w cn wit L S L S T T D n n T T D n n µ µ,.49 µ nd µ spctivly th lctonic nd ionic mobilitis. D nd D th dil diffusion cofficints of lctons nd ions. Ths fou pmts vy with th lctic fild, in pticul fo low vlus of lctonic nd ionic mobilitis Th lctonic mobility µ cn b vlutd oughly using Lontz fomul

17 µ 3 πmkt,.50 nσ wh m is th lcton mss, th lmnty lctic chg, n th dnsity of nutl gsous toms nd σ th coss sction of momntum tnsf. Fo ou clcultions, w hv howv usd BOLSIG simultions. Th sults dwn on figus nd nd ld fo Agon to th following djustmnt V µ P -8 P P..5 Th vlocity V is givn in m.s -, th lctic fild in V.m - nd th pssu P in b. Fig.. Vlocity V of lctons in Agon, clcultd using BOLSIG softw. 7

18 Fig.. Mobility of lctons in vious pu nd mixd gss, clcultd using BOLSIG softw. Concning th mobility µ of A + ions in Agon, xpimntl dt [9] psntd on figu 3. Th following fomul is obtind V µ n n n 75 n n,.5 wh V is givn in m.s -. n is th dnsity of Agon toms. This djustmnt is vlid until /n. 0-8 V.m. 8

19 Fig. 3. Vlocity V of A + ions in Agon. Honbck [0] lso poposs dt fo H +, A + nd N + ions in thi spctiv gss lctonic nd ionic diffusion Lkshminsimh t l [3] giv svl xpimntl vlus fo th tio D / µ, wh D is th cofficint of diffusion ppndicul to th dift diction of lctons. Thy hv shown tht this tio is bout constnt nd is woth 7 Volt fo /n vlus low thn 000 Td. Th losss ssocitd with this diffusion mod will not b tkn into ccount in this study. Concning th diffusion of th chgs in th diction of thi dift, fw xpimntl o thoticl dt xist. obtson t l [] popos som vlus fo th Agon ions, howv thy min thoticl, D bing too smll to b msud ccutly. Nvthlss, th cofficints D nd D cn b vlutd thnks to th instin fomuls D D kt µ ktµ ktµ..53 Ions hvy nd collid fquntly with th nutl toms. Thus, it is commonly dmittd tht th ionic tmptu T is constnt, clos to th tmptu T of th gs. Th lctonic tmptu T cn b clcultd using BOLSIG softw s figu 4. W obtin fo xmpl D D 0. m /s,.54 m /s 9

20 fo Agon t pssu of on b, tmptu T of 300 K nd n lctic fild of 0 6 V.m -. Fig. 4. lctonic tmptu T in Agon s function of th lctic fild, clcultd using BOLSIG softw. 3. Appliction to sub-minitu cylindicl fission chmbs Figu 5 shows two xpimntl clibtion cuvs obtind with CFUT-C3 chmb t ILL fo two nuton fluxs. At odiny fission ts 9 f/s, th clibtion cuvs psnt stution zon, wh th cunt I dlivd by th fission chmb vis littl with th voltg V pplid to th lctods. Th I st cunt obtind t th stution pltu is clld th stution cunt. But, t high fission ts high thn f/s, w obsv dfomtion of th stution pltu which nds to vnish compltly. Consquntly, th I st cunt cn not b vlutd ny mo with ccucy. 0

21 Fig. 5. xpimntl clibtion cuvs obtind with CFUT-C3 fission chmb t ILL/Gnobl High-Flux cto. Th stution pltu is dfomd t high fission ts nd th cosponding stution cunt cn not b dtmind unmbiguously. In this sction, w will gdully pply th thoticl fmwok dscibd in th pvious pgs in od to clcult nd to study th clibtion cuvs of sub-minitu cylindicl fission chmbs iditd in nuton fluxs vying fom low to high intnsitis. 3.. Cunt nd chg dnsitis t th stution zon At odiny fission ts, th I st cunt is on of th cucil infomtion dlivd by fission chmb opting in cunt mod. W will consquntly psnt in this sction th influnc on th I st vlu of svl pmts such s th lngth of th Unium dposit o th diuss of th lctods Stution cunt, / ppoximtion of N nd ssocitd o At th stution pltu, th combintion nd vlnch tms ngligibl d. W cn thus wit nv N. 3. n v N xpssions.7 nd.4 thn nbl us to clcult th stution cunt I I π h N d. 3. st d In fct, th coct fomultion would b th tht th pltu xists bcus th combintion nd vlnch pocsss ngligibl.

22 If w ppoch N by th / xpssion givn by., w obtin I st π hx N X τ fst 0 f This common ppoximtion inducs dictly n o on th stution cunt tht w wnt to stimt. W not ε, th o fcto givn by I st clcultd fo th/ ppoximtion of N ε. 3.4 I clcultd using xct xpssion of N st W cn vlut ε fo vious coupls {, } s tbl. [mm] [mm],5,5,75 0,43 0,48 0,5 0,55,5 0,4 0,47 0,5,5 0,4 0,46,75 0,39 Tb.. Vlus of ε fo vious {, } coupls. h cm. W consquntly notic tht th o md on I st by th / ppoximtion is impotnt lmost fcto fo fission chmbs of millimt sizs. It dcss gdully whn th cthod dius incss volution of I st cunt with h, th dposit lngth W wnt to stimt in this pgph th impct on th stution cunt of modifiction of th dposit lngth h. xcptionlly, w will us fo this study th xct xpssion of N,z, th dnsity of pis ctd by th fission poducts p scond. This dnsity is givn by th.4 qution. At stution, th combintion nd vlnch phnomn scondy nd ll th pimy chgs ctd collctd. W cn thus wit I st Ω N, z ddθdz, 3.5 wh Ω is th int-lctod volum. Using.8 sult, w consquntly vify tht th o on I st du to.7 ppoximtion is null. xpssion 3.5 cn b wittn in th following wy I st X 0τ f G h,,, zmin, zmx, 3.6 π with G h φ φ ccos / z z z' h / + cos + z z' ccos / z mx min h / dz' dzdφd. 3.7

23 3 It cn thus b bokn up into two contibutions: I I + I, givn by +, /,,, /,,,, /, /,,, mx min 0 0 z h h G h z h G X I h h h G X I f f π τ π τ. 3.8 Th I nd I contibutions spctivly th cunts ctd insid nd outsid th dposit zon. Fo CFUT-C3 chmb, w hv.5 mm,.75 mm, z min cm nd z mx 0.7 cm. Whil vying h, w obtin th figus 6 nd 7. Fig. 6. tios in pcntg of th I cunt ctd outsid th dposit zon ov th totl cunt I st, s function of th dposit lngth h z min cm, z mx 0.7 cm,.5 mm,.75 mm.

24 Fig. 7. volution of th fcto G with th dposit lngth h z min cm, z mx 0.7 cm,.5 mm,.75 mm. Fist, w obsv tht th fction of cunt ctd outsid th dposit zon is wk fo usul lngths h, clos to cm, s th I /I tio is 6.8 %. Scondly, th totl cunt dlivd by th chmb vis vy littl with th siz h of th dposit. Btwn h mm nd h cm, th vition on I st is low thn % volution of th stution cunt with th diuss of lctods Figus 8 to psnt th volution of th G fcto, intoducd in th pvious pgph nd givn by qution 3.7, s function of th nd diuss of nod nd cthod. 4

25 Fig. 8. Abcus of G vlus fo vious {, } coupls h cm, z min cm, z mx 0.7 cm. Fig. 9. volution of th fcto G with th int-lctod volum V z mx - z min - h cm, z min cm, z mx 0.7 cm. Fig. 0. volution of th fcto G with th int-lctod gp d - h cm, z min cm, z mx 0.7 cm. 5

26 Fig.. volution of th fcto G with th nod dius.75 mm, h cm, z min cm, z mx 0.7 cm. W notic tht, conty to intuition, G is not popotionl with th volum of th intlctod zon, which is popotionl to -, but th with th int-lctod gp, i.. with Compison btwn thoticl nd xpimntl vlus of I st Fo stndd CFUT-C3 clss chmb, th fcto G is mm s figu 8. Using th vlu X P [b] pis.m - s sction., w hv consquntly I st X τ 0 f -4 G P[b] π τ f. 3.9 Duing th pst th ys, w hv iditd t svl occsions CFUT-C3 fission chmbs in th High Flux cto HF of Lu Lngvin Institut ILL [,6]. Th Agon pssus insid th chmbs ngd btwn 0.9 nd.6 b, th nuton fluxs btwn nd.0 5 n.cm -.s - nd th 35 U dposit msss btwn 4 nd 40 µg. Th tios of th clcultd cunts ov th xpimntl ons plottd on figu. 6

27 7 Fig.. tios of th thoticl cunts ov th xpimntl ons s function of th fission t. Th xpimntl vlus hv bn obtind t th opting point of th clibtion cuvs s sction W obsv n ovll good gmnt btwn xpimntl dt nd thoticl clcultions fo fission ts low thn f/s. 3.. Distotion of th lctic fild t stution In cylindicl gomty, th lctic fild obys qution.45. In bsnc of lctic chgs, is ducd to xpssion.48. As consqunc, w cn wit s th sum of two contibutions δ + chgs without, 3.0 with ln ln ln 0 0 chgs without d n n d n n V ε ε δ. 3. W cn stimt th dnsitis n nd n t stution using 3. systm d N v n d N v n. 3.

28 8 By -injcting ths xpssions in 3. qutions, w obtin ln ln 0 chgs ithout w d d N v d N v d d N v d N v V ε δ. 3.3 At th fist od, whn th ptubtions du to th spc chgs not too impotnt nd by nglcting th diffusion pocsss, th lctonic nd ionic dift vlocitis givn by V v V v w w w w chgs ithout chgs ithout chgs ithout chgs ithout ln ln µ µ µ µ. 3.4 In th following clcultions, th µ nd µ mobilitis will oby.5 nd.5 xpssions. In ddition, s s qutions. nd F h X F X N N f fst π τ π, 3.5 w cn not tht f τ δ Th high th fission t, th mo th ptubtions du to th spc chgs bcom impotnt.

29 Fig. 3. Distotion of th lctic fild s function of th dil distnc fo svl voltgs V th filling gs is Agon t pssu of b,.5 mm,.75 mm, h cm, τ f 0 9 f/s. Fig. 4. Distotion of th lctic fild s function of th dil distnc fo svl voltgs V th filling gs is Agon t pssu of b,.5 mm,.75 mm, h cm, τ f 0 9 f/s. W notic on figus 3 nd 4 tht th spc chgs induc diminution of th lctic fild t nod nd n incs t cthod. This fild distotion ffct is infocd whn th voltg dcss. Indd, t low voltgs, th lctic chgs mov slowly. Consquntly, thi collction tims incs nd thy ccumult in gs, ptubing th lctic fild. W lso obsv tht th distotion is mximl t cthod: δ/ mx δ/, nd flls quickly whn th int-lctod gp dcss. 9

30 Fig. 5. Mximl distotion of th lctic fild s function of th voltg V th filling gs is Agon t pssu of b,.75 mm, h cm, τ f 0 9 f/s. Fig. 6. Mximl distotion of th lctic fild s function of pssu P.75 mm, h cm, V 00 V, τ f 0 9 f/s. 30

31 Fig. 7. Mximl distotion of th lctic fild s function of th int-lctod gp d th filling gs is Agon t pssu of b,.75 mm, h cm, V 00 V, τ f 0 9 f/s. Figus 5, 6 nd 7 show spctivly tht δ/ mx vis in / V. δ/ mx vis in P. This sult, whos simplicity ws unxpctd, is minly du to th qusi-lin dpndncs in P of N, /µ nd /µ. δ/ mx vis lik d 5/4 t fixd dius fo d mm. This dpndnc shows us tht duction of fcto on th gp d involvs duction of fcto 3 on th spc chgs. It confims th sults shown in figus 3 nd 4. Consquntly, using 3.6 xpssion, w cn wit δ mx δ P τ d f K V 5 / 4, 3.7 wh th fcto K is dducd fom figus 3 nd 4. W obtin K V.s.b -.m -5/4 fo.5 mm nd K V.s.b -.m -5/4 fo.5 mm. W now bl to clcult th mximum fission t tht should not b xcdd in od to void spc chg ptubtions. W consid tht th spc chg ffct is ngligibl if δ mx 5 %. 3.8 Using fomul 3.7, this condition is quivlnt to 3

32 V τ f 0, / 4 K P d Fo stndd CFUT-C3 clss chmb,.5 mm nd.75 mm. At P b, w obtin thus τ f f/s fo V 00 V. τ f f/s fo V 00 V. τ f f/s fo V 300 V. Fo CFUT-C6 clss chmb,.5 mm,.75 mm nd w hv t P b τ f f/s fo V 00 V. τ f f/s fo V 00 V. τ f f/s fo V 300 V. Consquntly, opting fission chmbs with ducd gps is n xcllnt mn to void spc chg ptubtions Simplifid clcultions of clibtion cuvs Whn th int-lctod gp is smll compd to, w cn modl th option of fission chmb with systm of qutions in pln gomty. If w nglct th diffusion pocsss nd th spc chgs phnomn, this systm cn b wittn nv N knn + αnv nv N knn + αnv V n 0, n Th stution cunt obtind with this ppoximtion is givn by I st S N. 3.. Th S.N fcto is vgd ov th int-lctod volum nd w hv S. N πh S N d N d. 3. Thus, it cn b chckd tht th stution cunt obtind with 3. ppoximtion is qul to th xct xpssion 3.. Th 3.0 systm cn b solvd nlyticlly nd w obtin 3

33 C Cµ Y Y + n tn k µ k µ µ, 3.3 with Y 4µ µ Nk C µ µ. C nd C two functions of tht oby th following systm C C µ µ µ µ C + ctn Y Y µ µ µ µ ctn Y Y C + α. 3.4 Th 3.3 qution thn nbls us to clcult th cunt I dlivd by th chmb ccoding to th voltg V pplid to lctods I πh V µ n, 3.5 Fo.5 mm,.75 mm, h cm, z min cm, z mx 0.7 cm, k m 3.s -, P b, X pis.m - nd τ f f/s, w obtin fo instnc th clibtion cuv I f V dwn on figu 8. Fig. 8. Clibtion cuv clcultd with 3.0 systm. Th filling gs is Agon. α is vlutd using Townsnd fomul. Th mobilitis µ nd µ givn by.5 nd.5 xpssions. Th cofficint α is givn by th.35 Townsnd fomul with Zstwny cofficints [] 33

34 5 A.0 0 P 5 B P [b] [b]. 3.6 This fomul is simplistic nd dos not poduc pcisly th xpimntl dt fo wk lctic fild. As consqunc, w xpct poo qulity dsciption of th bginning of th vlnch. Nvthlss, w notic on figu 8 tht th shp of th clibtion cuvs is coctly poducd by th modl, with cl stution zon fmd by th combintion mod on th lft nd th vlnch mod on th ight. With 3.3 nd 3.5 xpssions, w cn now study th dfomtion of clibtion cuv ccoding to pmts such s th gs pssu, th fission t o th int-lctod gp volution of th shp of clibtion cuv with th combintion cofficint k Whn th cofficint k dcss, w obsv on figu 9 n incs in th siz nd qulity of th stution pltu. Fig. 9. Thoticl clibtion cuvs fo vious cofficints k th filling gs is Agon t pssu of b,.5 mm,.75 mm, τ f f/s, X pis.m -. α is vlutd using Townsnd fomul. On figus 30 nd 3, w hv plottd th volution of V min, V 0, V mx nd I 0 s function of th combintion cofficint k. V 0 is th voltg obtind t th inflction point of th clibtion cuv nd I 0 th cosponding cunt. V 0 nd I 0 spctivly clld th opting point nd opting cunt. V min nd V mx th voltgs t th fist nd lst point of th stution pltu, spctivly. Thy dfind fo I min 0.98 I 0 nd I mx.0 I 0. 34

35 Fig. 30. volution of V min, V 0 nd V mx with cofficint k th filling gs is Agon t pssu of b,.5 mm,.75 mm, τ f f/s, X pis.m -. α is vlutd using Townsnd fomul. Fig. 3. volution of I 0 with cofficint k th filling gs is Agon t pssu of b,.5 mm,.75 mm, τ f f/s, X pis.m -. α is vlutd using Townsnd fomul. Fist, w notic tht th V mx voltg nd th I 0 cunt lmost indpndnt of k on two ods of mgnitud. This sult is not supising, s th combintion pocsss ngligibl t stution s sction.5 o 3.. Scondly, whn th cofficint k incss, th siz of th stution pltu, givn by th diffnc btwn V mx nd V min, is ducd. Indd, whn k bcoms impotnt, th combintion pocsss min non-ngligibl vn t high voltgs nd th tnsition with th stution mod is consquntly dlyd volution of th shp of clibtion cuv with th fission t 35

36 As shown on figus 3 nd 33, whn th fission t incss, th stution pltu gdully dispps nd th tnsition btwn th combintion nd th vlnch mods is lss nd lss visibl. Th stution pltu is pogssivly plcd by mixing of combintion nd scondy chg poduction. This bhviou is obsvd xpimntlly s ldy mntiond in figu 5. Fig. 3 nd 3b. Pogssiv dfomtion of th clibtion cuvs t high fission ts th filling gs is Agon t pssu of b,.5 mm,.75 mm, k m 3 /s, X pis.m -. α is vlutd using Townsnd fomul. 36

37 Fig. 33. Divtivs of th clibtion cuvs fo diffnt fission ts th filling gs is Agon t pssu of b,.5 mm,.75 mm, k m 3 /s, X pis.m -. α is vlutd using Townsnd fomul. In ddition, w notic on figu 34 tht th opting cunt I 0 dos not vy ny mo linly with τ f t high fission ts. This mjo ptubtion is pimily du to th infocmnt of th combintion tm kn n, th chg dnsitis incsing with τ f. Nvthlss, duction of th int-lctod volum must induc dcs in th numb of chgs psnt in th filling gs. As consqunc, w xpct tht th dvition fom th stndd lin lw will b ducd whn th int-lctod gp d dcss. This conclusion is vifid on figu

38 Fig. 34. volution of th opting cunt I 0 with th fission t τ f fo two nod diuss th filling gs is Agon t pssu of b,.75 mm, k m 3 /s, X pis.m -. α is vlutd using Townsnd fomul. Fig. 35. Dvition in pcntg fom th stndd lin lw s function of th fission t. I th would hv bn th cunt xpctd if th lin lw ws tu th filling gs is Agon t pssu of b,.75 mm, k m 3 /s, X pis.m -. α is vlutd using Townsnd fomul volution of th shp of clibtion cuv with th gs pssu Whn th filling gs pssu P incss, th lctic chgs mov slow nd th numb of scondy ioniztions is consquntly ducd. To chiv vlnch, it is ncssy to tnsmit mo kintic ngy to lctons. An incs in P will thus induc shift of vlnch towds high voltgs, s illusttd on figu

39 Fig. 36. Thoticl clibtion cuvs fo vious gs pssus P th filling gs is Agon,.5 mm,.75 mm, τ f f/s, X P [b] pis.m -. α is vlutd using Townsnd fomul. Th dnsity N of lcton-ion pis ctd p scond by th fission poducts incss linly with th pssu P vi th X 0 cofficint. Moov, th I st cunt is indpndnt of th mobilitis nd of th fist ioniztion cofficint. Consquntly, s w cn notic on figu 37, I 0 is qusi-lin function of P fo pssus clos to th tmosph. Fig. 37. volution of I 0 with th pssu P th filling gs is Agon,.5 mm,.75 mm, τ f f/s, X P [b] pis.m -. α is vlutd using Townsnd fomul. 39

40 Stting fom now, w will vy th pssu P whil dpting th fission t τ f in od to mintin th stution cunt I st constnt. In gmnt with 3.9 qution, fixing I st is quivlnt to fix th poduct P τ f. Fig. 38. Thoticl clibtion cuvs fo vious gs pssus P. Th fission t is djustd in od to mintin th stution cunt constnt th filling gs is Agon,.5 mm,.75 mm, X P [b] pis.m -. α is vlutd using Townsnd fomul. W notic thn on figu 38 tht n incs in P inducs diltion of th clibtion cuvs. In pticul, th siz of th stution mod is incsd s figu 39. As consqunc, woking t high pssus fcilitts th msumnt of th stution cunt, nd thus th xpimntl dtmintion of th fission t. But w hv to mmb tht th spc chgs ptubtions lso incs with th pssu s P s chpt

41 Fig. 39. volution of V min, V 0 nd V mx with th pssu P th filling gs is Agon,.5 mm,.75 mm, τ f f/s, X P [b] pis.m -. α is vlutd using Townsnd fomul volution of th shp of clibtion cuv with, th nod dius At fixd cthod dius, n incs in inducs infocmnt of th lctic fild, in ccodnc with.48 xpssion. Th chg vlocitis will consquntly gow nd lctons will induc li scondy ioniztions. Thus, th vlnch mod will stt t low voltgs, s illusttd on figu 40. W notic howv tht this dwbck is compnstd by n incs in qulity of th stution pltu. Indd, its slop pogssivly dcss whn th int-lctod gp d is ducd, llowing mo pcis dtmintion of th stution cunt. 4

42 4 Fig. 40. Thoticl clibtion cuvs fo vious nod diuss th filling gs is Agon t pssu of b,.75 mm, τ f f/s, X pis.m -. α is vlutd using Townsnd fomul. Th opting cunt I 0 mins clos to I st fo common fission ts infio to f/s fo CFUT-C3 chmb. Consquntly, its snsitivity to nd dii is confom to th sults obtind in sction Clcultion of th mximum voltg tht should not b xcdd to void vlnch In this sction, w will vlut fo vious pu nd mixd gss th mximum voltg V m tht should not b xcdd to void vlnch. At th nd of stution, i.. t th bginning of vlnch, th combintion pocsss ngligibl. Consquntly, th 3.0 systm bcoms , n n V v n N v n v n N v n α α. 3.7 W thus obtin [ ] N n α µ α, 3.8 nd th output cunt I is givn by

43 I α [ ] N π h. 3.9 α To stimt V m, w ssum th condition I V m.05 I st, 3.30 This condition sults in th following qution on m α α m m This qulity cn b solvd nd lds to condition on th fist ioniztion cofficint α α m. 3.3 Fo.5 mm nd.75 mm, w hv - α m 93.6 m Using BOLSIG simultions psntd figu 9, w hv solvd 3.33 qution fo svl gss. sults psntd in tbl 3. Gs m [0 6 V.m - ] Hlium 0.89 Non 0.89 Agon 0.50 A + X A + % X 0.39 Tb. 3. lctic filds t th bginning of vlnch in stndd CFUT-C3 chmb fo vious gss. With th 3.7 xpssion of th lctic fild, w finlly obtin Gs V m [Volt] Hlium 40 Non 40 Agon 50 A + X 30 A + % X 60 Tb. 4. Voltg t th bginning of vlnch in stndd CFUT-C3 chmb fo vious gss Fist, w obsv on tbl 4 tht Non o Hlium chmb cnnot wok t high voltgs, s vlnch pps too ly. Scondly, w notic tht th ddition of contminnts with low ioniztion potntils involvs shift of vlnch towds low voltgs, s th scondy 43

44 ioniztion pocsss fcilittd. In pticul, Xnon contmintion cn hv ctstophic consquncs on th siz of th stution pltu. It is thus impotnt to mmb tht this gs is on of th most bundnt fission poducts of 35 U. Fo instnc, with 4 µg 35 U dposit in 0 5 n.cm -.s - thml nuton flux, th Xnon contmintion of Agon cn ch, ft 40 dys of idition, bout 0. % in stndd CFUT-C3 chmb t pssu of on b. Th Xnon ffct is illusttd on th xpimntl clibtion cuvs psntd in figu 4, wh w obsv shift of vlnch towds low voltgs in th cous of idition Fig. 4. volution of th shps of xpimntl clibtion cuvs duing idition t ILL/Gnobl High-Flux cto in 0 5 n.cm -.s - thml nuton flux CFUT-C3 chmb with 98.5 % pu 35 U dposit. 4. Conclusion In this ticl, w hv poposd n ovviw of th physicl pocsss tht tk plc insid fission chmbs woking in cunt mod. W hv lso dvlopd n nlyticl modl llowing th clcultion of thi chctistic sponss ccoding to pmts such s th gs pssu o th fission t. Th shps of ths chctistic cuvs hv bn coctly poducd nd thi volution cosponds to th xpinc fdbck ccumultd duing 5 ys t ILL\Gnobl High Flux cto. Th clcultion of th lcton-ion pis dnsity ctd in th filling gs by th fission poducts ws cid out. As consqunc, th pdiction of th cunt dlivd t stution ws chivd with pcision. In ddition, th spc chg ptubtions hv bn quntifid nd th clcultion of th mximum voltg tht should not b xcdd to void vlnch hs bn conductd. W pdict in pticul shift of vlnch towds low voltgs whn contminnts such s Xnon fission poduct of Unium 35 psnt in Agon gs. This ptubtion is vifid duing xpimnts nd complicts notbly th post-idition nlysis. Finlly, w hv noticd on svl occsions tht duction of th int-lctod gp fcilitts th msumnts in high nuton fluxs whil limiting th ptubtions usully ssocitd with high fission ts spc chgs ptubtions, dfomtion of th cuvs, disppnc of th pltu, tc.. 44

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