The Fourier Transform Solution for the Green s Function of Monoenergetic Neutron Transport Theory

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1 4 Hawaii Univrsity Intrnationa Confrncs Scinc Tchnoogy Enginring ath & Education Jun 6 7 & 8 4 Aa oana Hot Honouu Hawaii Th Fourir Transform Soution for th Grn s Function of ononrgtic Nutron Transport Thory Ganapo Barry D. Univrsity of Arizona Dpartmnt of Arospac and chanica Enginring

2 Ganapo Barry D. Dpartmnt of Arospac and chanica Enginring Univrsity of Arizona Th Fourir Transform Soution for th Grn s Function of ononrgtic Nutron Transport Thory Nary 65 yars ago Kn Cas pubishd his smina papr on th singuar ignfunction soution for th Grn s function of th mononrgtic nutron transport quation isotropic scattring. Prviousy th soution had bn obtaind by Fourir transform. Whi it is apparnt th two had to b quivant a convincing quivanc proof for gnra anisotropic scattring rmaind a chang unti now. Th Fourir Transform Soution for th Grn s Function of ononrgtic Nutron Transport Thory INTRODUCTION It has bn nary 55 yars sinc th drivation of th most maningfu anaytica soution of nutron transport thory. Kn Cas in a rmarkaby insightfu papr [] appid sparation of variabs to xprss th Grn s function for mononrgtic isotropicay scattring nutrons in trms of singuar ignfunctions. Prviousy th soution had bn found anayticay through Fourir transform invrsion [34]. Hr w wi addrss th qustion of quivanc btwn th two approachs. Showing quivanc for othr than isotropic scattring is anything but straightforward (as shown by th author in [5]. Whi quivanc was indd dmonstratd it ackd intuitiv simpicity. In th foowing w rvisit quivanc in a mor unifid mannr through th gndr poynomia xpansion of th Fourir transformd soution in trms of anayticay dtrmind momnts.. Th ononrgtic Grn s Function Our focus is th Grn s function givn by th nutron transport quation c ψ ( x ; ω P( ψ ( x; δ ( δ( x x = = satisfying th condition im ψ ( x ; x (a <. By transationa invarianc th sourc mitting nutrons in dirction is ocatd at x =. Th tota cross sction is unity and c is th numbr of scattring scondaris [ c < ]. ω is th th scattring cofficint for a gndr poynomia [ P ( ] sris xpansion of th scattring krn. Th gndr momnts

3 ( ( ( ψ x; d P ψ x ; ; =... (b wi pay a ky ro in what foows. As usua and x ar th nutron dirction and position of ψ x ;. th anguar fux distribution (. Th Standard Soution Th anguar fux soution drivs from th Fourir transform of Eq(a and th gndr momnts to giv ikx ( k ; dx ( x ; ψ ψ (a ikx ( k; dk ( x; ψ ψ (b c ; ;. = ( ik ψ ( k = ω P ( ψ ( k δ ( (3 Soving for th anguar fux and projcting givs th momnts as th soution to whr th matrix mnt is xpicit z δ cω z ψ k = P ( ( ; ( (4 j j j = z ( ( ( ( j Q z P z j j ( z ( j z Q j z P z < j and z / ik. Q is th gndr function of th scond kind of ordr. Th soution to Eq(4 rquirs truncation of th scattring krn for which w hav assumd th scattring cofficint ω to vanish for. In vctor notation Eq(4 bcoms z I -c ( z W ( k; = P ( ψ (5 z

4 W { ;... } ( j( diag ( ω ( ( z z j= { Pj j } P ; ;... = and whos soution by matrix invrsion givs th anguar momnts vctor ψ z ( k; = I c ( z W P ( (6 z ( k; ( k; ( k;... ( k; ψ = ψ ψ ψ. Th anguar fux transform coms from Eq(3 ψ z c z ; δ ;. T ( k = ( P ( Wψ ( k From th invrsion intgra thrfor z z ψ ψ ikx ( x ; dk ( k ; π T x/ ψ = Θ δ ( x ; ( x/ ( c T z z P W 4π z z I W P ikx ( dk c ( z ( (7 whr ( x / Θ is th unit stp function in th uncoidd contribution which is th first trm. 3

5 Whi Eq(7 is a vaid soution an xpicit anaytica form from th invrsion is not forthcoming. On can ony say pos com from th zros of Dt c ( z points ±i com from th ogarithm in th gndr functions in. I W and branch 3. Th Non-Standard Soution 3. omnts Soution W find a mor usfu momnts soution by projction of Eq(3 ovr gndr poynomias to giv th foowing thrtrm rcurrnc for th transformd gndr momnts: ( ; ( ( ; ( ; ( zhψ k ψ k ψ k = zs (8 h ω S ( ( P (. From th thory of rcurrnc [6] th gnra soution to this rcurrnc is ( k; a( z; g ( z b( z; ( z z j( z S ( ψ = ρ α (9 j j= whr th functions g ( z and ( z ρ ar soutions to th homognous form of Eq(8 foowd by th particuar soution. Th cofficints a( z ; and b( z ; from th starting condition at = ( ; ψ ( k; ( S ( a z g ( z ρ ( z α ( z b z; = ; ; giving th rcurrncs ( ( ( ( ( ( ( ( zhg z g z g z zh ρ z ρ z ρ z. ar dtrmind Using th proprtis of th homognous soutions on can show th cofficint in th particuar soution is ( = ( ( ( ( zα z ρ z g z g z ρ z j j j 4

6 to giv ( k; g ( z ( k; ψ = ψ ( j g j( z ρ( z ρ j( z g( z Pj( j=. ( To conform to convntiona dfinitions of Chandraskhar poynomias t to giv for Eq( g ρ ( z ( g( z ( z ( ρ ( z ( k; = g ( z ( k; ( ψ ψ χ z (a χ ( z ( j P ( ( zg j ( z g ( z ( z. (b χ ρ ρ j j j= 3.. Transport cosur To continu w nd ( k ψ or in othr words cosur of th momnts quations [Eq(8]. ; To appy th xact transport cosur on sts j to zro in Eq(4 to giv on substitution of Eq(a ( ( ; ( δ cω z ψ k = = z ψ z ( k; = cz Q z z ( z ω z ( χ ( (3 Λ z = th disprsion ration aso writtn as ( z cz ω Q ( z g ( z Λ = 5

7 ( z ( g ( z Q ( z g ( z Q ( z Λ =. 3. Th Fourir Transform of th Anguar Fux W can convninty xprss th anguar fux transform as th gndr poynomia xpansion ( whr th momnts ( k = ( ( ( ψ k ; = ψ k; P (4 ψ ar from Eqs(. Th subscript indicats truncatd ; scattring. Whn th momnts ar introducd into Eq(4 k rpacd by k (for notationa convninc thr rsuts ( k ; ( z ( k; T( z ; ψ = φ ψ (5a ( ( ( ( (5b φ z g z P = and ( ; = ( ( ( T z χ z P. (5c Th sums in Eq(5bc rquir furthr carification whrby a gnraizd singuar ignfunction wi mrg. 3.. Gnraizd singuar ignfunctions To procd considr th imit ( N = aong th Christoff-Darboux formua N = ( N ( ( ( φ z = im g z P ( N gn ( z PN( gn ( z PN ( czg ( z g ( z P ( = z 6

8 whr g ( z ( ( ω g z P j j j j= g ( z For N on can show [ som ffort] and As a consqunc formay and =. ( ( z ( z ψ ( z ( z g z =Λ P Q ( z = ( g ( z P ( z g ( z P( z. ψ ( P ( z P( = ( z P ( P ( z P ( N PN N N N = im = δ N z = ( Q ( z P( = ( z P ( Q ( z P ( N QN N N N = im = N z to convninty giv (aftr som manipuation ( z cz g φ ( z = Λ( z δ ( z z ( z. (6a Simiary th scond kind poynomias gnrat th gnraizd singuar ignfunction 7

9 ( z = ( ( z P( Θ ρ or ( z z ch Θ ( z = γ ( z δ ( z z (6b whr Th xprssion δ ( z h j j j j= ( z ωρ ( z P (. ( z ( ( z Q ( z ( z Q ( z γ ρ ρ. rducs to th dta function and Eqs(6 bcom th Cas singuar ignfunctions on th branch cut in th compx z-pan as shown bow. 3.. Evauation of T( z ; With som manipuation th vauation of ( ; whr T z foows from Eqs(b and (5c z T( z ; = S ( z ; δ ( z δ ( z ( ; = Θ( φ ( Θ( φ ( ( ρ( z φ ( z φ ( z P ( = g ( z Θ( z Θ( z S z z z z z 3..3 Atrnativ xprssion for ψ ( Expoiting th foowing symmtry: on can show k; ( k ; ( k ; ψ ψ =. (7 8

10 ( k ( z ( z ( z γ ψ ; = φ Θ. (8 ( z Λ W ar now in position to invrt th transform of Eq(5a. 3.3 Fourir transform invrsion By introducing ψ ( k ; from Eq(8 into Eq(5a ( ; ψ ( bcoms k ; ( ( ( ( z ( z T z from Eq(7 γ ψ = φ φ (9 ( k ; z z H z ; Λ Θ H ( z ; P z δ ( z δ ( z ( z φ( z ( ρ( z φ ( z φ ( z (. = g ( z Θ ( z Θ ( z A brif dscription of th Fourir transform invrsion foows. Th pos and branch points of th intgrand ar and Fig. : ( Λ ± ν m = m =... ± i (from th ogarithm in th gndr functions rspctivy. Thn for th contour of ε i/ C Rε Γ R ε Γ R ε C Rε R Γ R C ε i i/ R 9

11 C Γ C Γ C Γ C ε ε ε ε ε ε Fig.. Compx k-pan: R R R R R R contributions from th po singuaritis and aong th branch cut giv whr th first trm is ( x ; I ( x ; I ( x ; ψ = (a Γ Γ ( ( φ ν m φ ν m I ( x ; m= x/ ν m =. (b ( ν Th scond trm in Eq(a is from th branch cut which is an intgration ovr th discontinuity of th boundary vaus of th sctionay anaytic intgrand and bcoms x/ ν i I ( x ; = dν Disc ψ k ( ν ; Γ Γ m ( π ν ( k ( ; k( Disc ψ ν ( ( k( ψ ν ; ψ ν ;. Th boundary vaus of ψ ar ( k( k( i ( ± ψ ν ; im ψ ν± ε ;. ε With som xtnsiv manipuation and noting H ( z ; branch cut contribution to driv from th first trm of Eq(5a to giv x/ ν I x d is anaytic aows th ntir ( ; = ν φ ( ν φ ( ν Γ Γ (c ( ν ( ( ( ν νλ ν Λ ν. ± Aso noting that th contributions from th contours C ε and vanish in thir rspctiv R imits Eqs( giv th fina soution for truncatd scattring for x as C ε

12 ( x ; ψ ( ν ( ν x/ ν m = m= φ φ m m ( ν m x/ ν dν φ φ ( ν ( ν ( ν. (a From rciprocity ( x ; = ( x ; ψ ψ and φ ( ν = φ ( ν Eq(a aso is ( x ; ( ν ( ν m = ψ m= φ φ m m x / ν ( ν ( ν m x / ν dν φ φ ( ν ( ν. (b Eiminating scattring truncation by tting givs th cassica singuar ignfunction xpansion for x as found by Cas whr < ( x ; ψ ( ± ν m ( ± ν m m = m= φ φ ( ν m x / ν ( ν x / ν dν φ φ ( ν c g ν ( ± ν ( ± ν ν * φ ( ν imφ ( ν = P Λ ( ν δ( ν = ( ν ω ( ν ( g g P (

13 Λ ( ν cν ωq( ν g( ν = and P is th principa vau of th xprssion foowing whn undr an intgra. FINA REARKS Th abov drivation mrits additiona commnt. Whi th rsut is not nw th stps gtting to it ar. Th primary rason th approach succds is that it is a consqunc of th soution to th momnts rcurrnc coming from an anaytica cosur. This is nov sinc an anaytica soution to a rcurrnc is not common in anaytica soutions to th transport quation. Rcurrncs gnray find us in numrica not thortica vauations. It is aso obvious that th approach works bcaus w know what to ook for th singuar ignfunction xpansion. Thus it is not too surprising that out Cas s guidanc th singuar ignfunction xpansion was not first discovrd from th Fourir transforms. Additionay no orthogonaity or comptnss is rquird as th ignfunction xpansion simpy mrgs from manipuation in th compx pan. For this rason th Fourir transform drivation whi ss gant than Cas s soution is mor cassicay mathmatica. REFERENCES. K.. Cas Emntary Soutions of th Transport Equation and Thir Appications. Ann. of Phys: 9-3 (96.. A.. Winbrg and E.P. Wignr Th Physica Thory of Nutron Chain Ractors Univrsity of Chicago. Prss Chicago ( B. Davison Nutron Transport Thory Oxford Prss ondon ( R.E. arshak Thory of th Sowing Down of Nutrons by Eastic Coisions Atomic Nuci Rv. od. Phys ( B.D. Ganapo A Consistnt Thory of Nutra Partic Transport in an Infinit dium Transport Thory and Stat. Phys (. 6. J. Wimp Computation rcurrnc rations Butin (Nw Sris of th Amrican athmatica Socity 4 no (986.

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