Calculation of Thermal Neutron Flux in Two. Dimensional Structures with Periodic Moderators

Transcription

1 Apple Mhemcl Scences Vol. 8 no Clculon of Theml Neuon Flu n Two mensonl Sucues wh Peoc Moeos S. N. Hossenmolgh S. A. Agh L. Monzen S. Bsn In Unvesy of Scence n Technology Tehn In epmen of Nucle Physcs hossenmolgh@us.c. In Unvesy of Scence n Technology Tehn In epmen of Nucle Physcs Absc In hs ppe we compue he heml neuon ffuson flu s funcon of spce n me n wo-mensonl sucues wh peoc moeos compose of ccul os wh sque lces. The moeos e ssume o be C n HO. Ou compuons e gven n wo mn seps. Sep one: he moeos nse n ouse of he o e ssume h become C n HO especvely.sep wo: hs sep s he nvese of sep one. In such sucues mpng consn egons e efne s egons n whch heml neuon ffuson cnno e plce ppe. We efne hese mpng consn egons s ffuson bn gps. Keywos : neuon moeo ffuson flu bn gp. Inoucon The fel of neuon physcs hs become n negl p of nvesgons no n y of mpon ssues h spn fels s vese s nucle n pcle physcs funmenl symmees sophyscs n cosmology funmenl consns gvon n he nepeon of qunum mechncs. We efne heml neuons o be neuons wh enegy nely 5mev.Fs neuons ech he heml egme mos effcenly hough csce of oughly collsons wh me ch n hyogen o eueum. Col neuons e pouce by cyogenc neuon moeo jcen o he eco coe o spllon ge hel empeue of K. One genelly wns he moeo s col s possble o ncese he phse spce ensy of he neuons. As he neuon wvelenghs become lge compe o he omc spcng he ol sceng coss secons n

2 9 S. N. Hossenmolgh S. A. Agh L. Monzen S. Bsn me e omne by elsc o quselsc pocesses n becomes moe ffcul fo he neuons o hemlze. Neuons e pouce fom fsson n esech eco n vege enegy of ppomely MeV. They e slowe o heml enegy n moeo such s hevy o lgh we gphe o beyllum suounng he fuel. Theml neuon ffuson hs ce much enon fo mny yes s one of he mos mpon poblems n nucle fsson [-] n fuson. In nucle fsson s mpon o conol heml neuon ffuson n nucle ecos. Fo emple he nhbon of heml neuon ffuson my pove novel pplcons n nucle ecos. In nucle fuson um whch s genee by he econ of lhum n heml neuons s neee. Theefoe he conol of heml neuons my lso pove novel pplcons n he geneon of um. Inee heml neuon ffuson s escbe by he ffuson equon..mhemclly howeve he ffuson equon wh cen mpng consn s nlogous o he wve equon wh cen fequency n heefoe sucues wh peoc moeos possess mpng consn egons n whch heml neuon ffuson cnno e plce le elecomgnec wves n phoonc bn gps.[ 7].In hs ppe we heoeclly emonse ffuson bn gps of heml neuon ffuson n heml neuon flu n wo-mensonl sucues wh peoc moeos compose of os wh sque lces. We o no conse heml neuon ffuson pepencul o wo-mensonl plnes. Moeos e ssume o be C n HO. ffuson bn gps sgnfcnly epen on he of os n he lce consns. Ths ppe s ve no 5 secons such h n secon I heoecl suggeson s gven n secon II we escbe he Foue nsfom of he ffuson coeffcen he bsopon coss-secon n he ffuson me especvely. In secon III by subsung hese Foue nsfoms no he neuon heml ffuson equon NTE wh some smplfcon we conve NTE o egenvlue poblem. In secon IV fom hs equon we eemne he egenvluesneuon mpng consns n egenfuncons neuon ffuson heml flu fo wo ffeen cses: cse moeos nse n ouse of he o e C n HO especvely cse moeos nse n ouse of he o e HO n C especvely. In fnl secon he obne esuls e gven.. Suggeson Theoy In fgue we show he schemc gm of wo mensonl sucues wh peoc moeos compose of os wh sque lces. []The egon embee by oe lnes nces he un cell. In fgue b we show he fs Blloun zone of sque lces wn by he she egon. As menone ele he fs Blloun zone s embee by he pepencul lnes whch ve equvlenly beween he ogn n he nees ecpocl pons o he ogn.the blc pons nce he nees ecpocl pons o he ogn. The oe lnes nce he pepencul lnes whch ve quvlenly beween he ogn n he nees

3 Clculon of heml neuon flu 9 ecpocl pons o he ogn n he Γ X n M pons e especvely. These pons nce hghly oonl symmec pons n he fs Blloun zone. Theefoe by he Bloch heoem wve vecos n gve he sme conon n we my nvesge only he fs Blloun zone n he wve veco spce. n Fgue. Schemc gm of wo-mensonl sucue wh peoc ffuson coeffcens. s he us of o s he lce consn of sque lces. b The fs Blloun zone of sque lces wn by he she egon. The blc pons nce he nees ecpocl pons o he ogn. The oe lnes nce he pepencul lnes whch ve equvlenly beween he ogn n he nees ecpocl pons o ogn n he Γ X n M pons e especvely. These pons nce symmec pons n he fs Bllon zone. Moeove ffuson bn sucues cn be obne by nvesgng wve vecos long he Γ X X M n M Γ segmens when sucues possess he C v symmey n he goup heoy. ffuson bn gps ppe wve vecos on oe lnes n fgue b. In he cse of unfom moeos ffuson bn gps o no ppe hese wve vecos. By openng of ffuson bn gps mpng consns ecese hese wve vecos whch cuses he ncese of elon me.e. he sblzon of he heml neuon flu. The econ of hs sble se nees cen eneges whch nces physcl menng of ffuson bn gps. Ths s sml o he cse of eleconc bn gps n sol ses [8-9].Any neuon souce cn be esgne o pouce lo of neuons. Howeve hgh neuon yel lone oes no epesen how useful hese neuons e. Flu s efne s he numbe of neuons cossng un e pe un me. I cn lso be

4 9 S. N. Hossenmolgh S. A. Agh L. Monzen S. Bsn vewe n he followng wy: nv n s he neuon ensy v s he velocy of neuon. A neuon souce wh hgh yel mus lso hve hgh neuon ensy n oe o pove hgh neuon flu. When he neuon geneo s couple wh neuon geneo he volume of he souce nceses. As esul he neuon ensy nceses slowe hn he yel n he neuon flu wll no be mpove lnely wh he ue sysem mulplcon fco.[-] In oe o obn ffuson bn sucues we s wh he ffuson equon fo heml neuon fo wo-mensonl sucues { } s ν T Whee ν T b ν n e he ffuson coeffcen he bsopon coss- T secon he heml velocy n he ffuson me especvely n n especvely. We o no conse he heml neuon souce s e he heml neuon flu n he heml neuon souce s n nvesgng popees of wo-mensonl sucues wh peoc moeos. vng hus: equon by [ ] { } The ffuson coeffcen he ffuson me he bsopon coss-secon n e peoc wh espec o he lce veco genee by he pmve nslon n hey my be epne n Foue sees on he ecpocl veco ep ep b Whee n ep c n e he Foue coeffcens of especvely. In wo-mensonl peoc sucues wh sque lces he lce veco s: l l Whee l n l e ny wo neges n bsc vecos e 5

5 Clculon of heml neuon flu 9 b 5 s he lce consn of sque lces. On he ohe hn he ecpocl lce veco s: 6 hb hb Whee h n h e ny wo neges n bsc vecos of hs ecpocl lce e: b 7 b 7b Then he lce veco n ecpocl lce veco ssfy he followng elon: 8 nege Theefoe he ffuson coeffcen he ffuson me n he bsopon coss- s escbe by secon e nslonlly symmec wh espec o. ep λ 9 In vng equon 9 usng sepon of vbles meho on he equon.whee λ s mpng consn. Usng Bloch s heoem [8-] we my epn s ep{ } Whee s Foue coeffcen of nces he econ of neuon ffuson. Then elon: ep. n s he wve veco whch ssfes he followng By ffeenon elve o me fom he boh se of equon 9 we obn: λ λ e Subsung eq. no we fn: [. { } ] λ O [. ] λ Usng equons b n we obn;[]. e 5 [ ]. e 6

6 9 S. N. Hossenmolgh S. A. Agh L. Monzen S. Bsn [ ] 7. e eplcng equons n -b no equon we obn: [ ] [ ]. ep [ ] [ ]. ep. [ ]. ep 8. e λ Now by mulplyng he boh se of bove elon on he e. fco n negon fom we wll hve: [ ] [ ]. [ ]. λ 9 Noe h n wng he bove equon we hve use he followng elon:[] { }. ep δ Whee s volume of un cell n δ s he Konee el funcon now usng he chnge of vble n he bove equon n we hve: [ ] λ [ ]. Also fo smplcy n hs equon we use he followng chnge of vbles: n heefoe he bove equon s convee o: K λ [ ]. K

7 Clculon of heml neuon flu 95 O λ K. K An: {. } λ In he fnl fom bove elon cn be wen n he followng m egenvlue poblem: λ Whee Φ 5 { } 6 Fom hs elon we see h fo fe wve veco of we wll hve one goup of soluons n fs Blloun zone hus by chngng wve veco of we hve nohe goup of soluons h e gven by ne n [9] whee hs ne epesene bn sucue.the eque conon fo hvng gp beween bns λ s [9]:.We cn nvesge hs conon n he followng fom: λ Fom equon 5 we hve: o hus by he mnus sgn n hs epesson some smplfcon we fn epesene h econ neuon wve veco s oppose he econ of ecpocl lce veco.e by ncesng he vlue of lce veco neuon wve veco s ecese.theefoe hs conon s ssfe n concluson ceon of he bn gp sucue s epece. In epesson 6 he m fom of cn be evlue n he followng meho: Clculon of Foue coeffcen of We now h: ep ou ou n n ou ou ep ep ep n n ep 7 ou ou

8 96 S. N. Hossenmolgh S. A. Agh L. Monzen S. Bsn nces egon nse o. A The fome negl n equon 7 ep δ. 8 Whee δ s he Konece el. B The le negl n equon 7 ep ep 9 θ J l ep lθ l θ ep snθ J Whee J l s he l h-oe Bessel funcon. We use he followng elon[-]: ep { snθ } J ep{ lθ} l By usng he elon: { zj z } zj z z We cn obn: ep l J coeffcen of s gven by: ou ou n n Theefoe he Foue ou ou f J ou ou n n l f whee f.in he sml wy n e gven by : ou n ou { } f J ou n 5 f

9 Clculon of heml neuon flu 97 n { } { } 6 n ou n ou ou f J f The m fom of n e gven n he followng fom especvely: 7 Whee { } ou n ou 7 { } b J ou n 7 { } c J ou n 7 An: 8 Whee: { } ou n ou 8 { } b J ou n 8 { } c J ou n 8 Also;

10 98 S. N. Hossenmolgh S. A. Agh L. Monzen S. Bsn 9 Whee ou ou n n ou ou 9 J n n ou ou 9 b { } J 9 n n ou ou { c. Numecl compuons: Ths secon s ve o wo mn ps: - eemnon of heml neuon flu wh conseng he moeos nse n ouse of he o s C n HO especvely. By nvesng hese mces we wll eemne he m fom of : Fo emple he fs elemen of m by he followng meho s eemne: '' We ssume h pons e he nees ecpocl pons o he ogn such h he componens e gven by :.Theefoe n n. n po po K K po The eque numecl vlues fo ffeen moeos C n H o n ou { } c cm C c.. compuons e gven by []: 8 cm. 7s n H O H o. 6cm. 97. s especvely. Une H o

11 Clculon of heml neuon flu 99 hese conons numecl vlue of m fom of n he econ of ΓX s gven by: 6.76e -.9e e - 6.9e -.9e - 6.7e -.9e e e - 6.9e - 6.7e -.9e -.9e e - 6.9e - 6.7e - Also he egenvecos of m e gven n he followng: The gonl fom of m s gven by: 9.67e - 7.e - 8.6e - 7.e - Noe h he gonl elemens of hs m e he egenvlue of m.we conclue h he neuon mpng consns h e ele o bn gp sucue e gven by: λ.67 λ 7. λ 8.6 n λ 7.. Now f we nse he egenfuncons of heml neuon flu no he Foue v nsfoms elon: j ep j. he pons: j.we fn:... e e e. e.... e e e e.... e e e e.... e e e e 5 Combnng eq. n 5 hus fnlly he obne esuls fo fou funcons s funcon of spce e gven n he below especvely: y e.5e.5.5e 5 y

12 9 S. N. Hossenmolgh S. A. Agh L. Monzen S. Bsn y e.e.77.e 5 b y y y e.5e.99.5e 5 c y e.77e.77e 5 Fgues - o -show h he vons of y n s funcon of n y.fom hese fgues we see h heml neuon flu n hve ffeen pe pons lso n ffeen pons e negve n hee s no ccepble snce hs egon s nown s bn gp. Noe h fom Fg-we fn h some pons wh coones y he neuon heml flu s posve n ccepble n he ohe pons he numecl vlues of e negve h s nown s bn gp egon.[-5] Thus n ll pons of y e ccepble. Fg.-: The vons of cm s funcon of n y cm 5cm /. 5 when he moeos nse n ouse of he o e C n HO especvely. s Fg. -b: The vons of cm s funcon of n y 5cm /. 5 when he moeos nse n ouse of he o e C n HO especvely. s cm

13 Clculon of heml neuon flu 9 Fg.-c: The vons of cm s funcon of n 5cm /. 5 when he moeos nse n ouse of he o e C n HO especvely. s y cm Fg.-: The vons of cm s funcon of n y 5cm /. 5 when he moeos nse n ouse of he o e C n HO especvely. s cm Now by usng he elon ep λ we cn eemne he me epenence of n he obne esuls e gven n he fgues - o -. Noe h n ou compuons we e one of he pe pons on fgues - o - fo emple 5 n 5 especvely.fom hese fgues we see h by ncesng me he heml neuon flu s ecese n he numecl vlues of hese funcons epen on he pe pons songly. Fg-: The vons of me epenence of pe pon n 5cm /. 5 when he moeos nse n ouse of he o e C n HO especvely.

14 9 S. N. Hossenmolgh S. A. Agh L. Monzen S. Bsn Fg-b: The vons of me epenence of 5cm /. 5 n when he moeos nse n ouse of he o e C n HO especvely. Fg-c: The vons of me epenence of 5cm /. 5 n 5 when he moeos nse n ouse of he o e C n HO especvely. Fg-: The vons of me epenence of 5cm /. 5 n 5 when he moeos nse n ouse of he o e C n HO especvely. Also he numecl vlues of vesus me e gven n he Tbles. Tble : numecl vlues of n vesus me efne pons 5 n 5 especvely 5cm /. 5 when he moeos nse n ouse of he o e C n HO especvely

15 Clculon of heml neuon flu 9 Fom hs ble we conclue h e no ccepble vlue becuse he pe pon 5 e plce n bn gp egon. -b eemnon of heml neuon flu wh conseng he moeos nse n ouse of he o s HO n C especvely. In hs p n sml wy he ol compuons of p e cheve ecep h we echnge he moeo nse of he o wh HO n ouse wh C. Theefoe he obne esuls e gven n he below.in hs cse we fn An s egenvecos e gven by The gonl fom of s gven by Such h. e. e. e e..... e e e e.... e e e e e e e e 9 Also pons.77.e.77 5 we fn:

16 9 S. N. Hossenmolgh S. A. Agh L. Monzen S. Bsn y y.5e.5e.99.5e 5b y.5e.5e.5.5e 5c y e.77e.77e 5 Fgues - o -show h he vons of n s funcon of & y.fom hese fgues we see h heml neuon flu n hve ffeen pe pons only n ll ffeen pons e posve n ccepble bu n n ll pons e negve n e no ccepble snce hese egons conn bn gp sucue. AlsoFg.- shows h n some pons e negve n ohe pons e posve n ccepble. y y Fg.-: The vons of cm /. 5 especvely. s cm s funcon of n y cm when he moeos nse n ouse of he o e HO n C Fg. -b: The vons of cm s s funcon of n y cm cm /. 5 when he moeos nse n ouse of he o e HO n C especvely.

17 Clculon of heml neuon flu 95 Fg.-c: The vons of cm /. 5 especvely. s cm s funcon of n y cm when he moeos nse n ouse of he o e HO n C Fg.-: The vons of cm /. 5 especvely. s cm s funcon of n y cm when he moeos nse n ouse of he o e HO n C Now by usng he elon ep λ we cn eemne he me epenence of n he obne esuls e gven n he fgues 5- o 5-. Noe h n ou compuons we e one of he pe pons on fgues 5- o 5- fo emple 5 n 5 especvely.fom hese fgues we see h s no ccepble pon n by ncesng me e ecese pon 5 n 5. Fg5-: he vons of me epenence of cm /. 5 n when he moeos nse n ouse of he o e HO n C especvely.

18 96 S. N. Hossenmolgh S. A. Agh L. Monzen S. Bsn Fg5-b: he vons of me epenence of cm /. 5 n when he moeos nse n ouse of he o e HO n C especvely Fg5-c: The vons of me epenence of cm /. 5 n 5 when he moeos nse n ouse of he o e HO n C especvely. Fg5-: The vons of me epenence of cm /. 5 n 5 when he moeos nse n ouse of he o e HO n C especvely. Also he numecl vlues of vesus me e gven n he Tbles. Tble:Numecl vlues of n vesus me efne pons 5 n 5 especvely cm /. 5 when he moeos nse n ouse of he o e HO n C especvely s Conclusons In mels connng oms of low omc mss neuons of ll eneges cn lose sgnfcn fcon of he enegy n sngle elsc collson n such

19 Clculon of heml neuon flu 97 mels e efee o s moeos. In hevy nucle ppecble enegy loss n collson s only possble hgh eneges whee nelsc sceng cn occu. We obn fom ou clculons hee mjo esuls such s :Fom ou gven suggeson heoy we cn conve he neuon heml ffuson equon o solvble egnvlue poblem by usng Foue nsfom echnque. We cn ble o clcule he heml neuon flu n neuon mpng consns n fs Blloun zone by usng he Mhemc Pogm n we fn h he heml neuon flu on hve connues sbuons n un cell..by nlyzng he suggeson heoy n scusson on he obne esuls we cn fn he physcl chcescs of bn gp sucues n wo followng seps: Sep one: when he moeos nse n ouse of he o e C n HO especvely. Sep wo: when he moeos nse n ouse of he o e HO n C especvely. In such sucues ffuson bn gps show h heml neuon ffuson cn no e plce ppe n mpng consns. ffuson bn gps ppe when C n HO e nse n ouse he os n vce ves. By compng he obne esuls fom sep one n wo we fn h he neuon mpng consns n heml neuon flu e no he sme becuse hese pmees e epen on he lce consn us of he o n choce of he suble moeos. Acnowlegmen Thns e ue o esech Councl of In Unvesy of Scence & Technology fo fnncl suppo of hs esech. efeences [] J.. Lmsh Inoucon o Nucle eco Theoy eng MA: Ason-Wesley 966. [] J.. Lmsh n A. B Inoucon o Nucle Engneeng Eon. []. ozon Inoucon o eco Knecs Polyechnque Inenonl Pess Monel 998. [] T. Hoyu n Y. Ksum J. Phys. : Appl. Phys. 6. [5] S. John Phys. ev. Le [6] E.Yblonovch Phys. ev. Le [7] H. Hym T. Hmno n Y. Aoyg Appl. Phys. Le

20 98 S. N. Hossenmolgh S. A. Agh L. Monzen S. Bsn [8] C. Kel Inoucon o Sol Se Physcs 6h en New Yo: Wley 986. [9] N.W. Ashcf n N.. Memn Sol Se Physcs. Hol neh n Wnson New Yo 976. [] E. Ks Aomc n Eleconc Sucue of Sols Cmbge Unvesy Pess. []. useppe n P.P. useppe Sol Se Physcs. Acemc Pess Sn ego Clf.. []. Afen n H. Webe Mhemcl Mehos fo Physcss 5h eon Acemc Pess Sn ego. [] C. Hpe Anlyc Mehos n Physcs. Wley-VCH Beln 999. []. Hempelmnn Quselsc Neuon Sceng n Sol Se ffuson Ofo Unv. Pess. [5]. L. Sques Inoucon o he Theoy of Theml Neuon Sceng ove Pubns 997. eceve: ecembe 8 7