PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 CH.VII: NEUTRON SLOWING DOWN INTRODUCTION SLOWING DOWN VIA ELASTIC SCATTERING • • • • KINEMATICS SCATTERING LAW LETHARGY DIFFERENTIAL CROSS SECTIONS SLOWING-DOWN EQUATION • P1 APPROXIMATION • SLOWING-DOWN DENSITY • INFINITE HOMOGENEOUS MEDIA SLOWING DOWN IN HYDROGEN • HYPOTHESES • FLUX SHAPE • SLOWING-DOWN DENSITY SHAPE 1 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 OTHER MODERATORS • PLACZEK FUNCTION • SYNTHETIC SLOWING-DOWN KERNELS SPATIAL DEPENDENCE • FERMI’S AGE THEORY • SLOWING DOWN IN HYDROGEN 2 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 VII.1 INTRODUCTION Decrease of the n energy from Efission to Eth due to possibly both elastic and inelastic collisions Inelastic collisions: E of the incident n > 1st excitation level of the nucleus • 1st excited state for light nuclei: 1 MeV • 1st excited state for heavy nuclei: 0.1 MeV Inelastic collisions mainly with heavy nuclei… but for values of E > resonance domain Elastic collisions: not efficient with heavy nuclei With light nuclei (moderators) Objective of this chapter: study of the n slowing down via elastic scattering with nuclei of mass A, in the resonance domain, to feed a multi-group diffusion model (see chap. IV) in groups of lower energy 3 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 VII.2 SLOWING DOWN VIA ELASTIC SCATTERING KINEMATICS Absolute coordinates of n Before collision v v E’ E Deflection angle: o '. n vr ' v' Before collision After collision v ' v' ' G A G vG v c.o.m. system After collision 1 vG v' ' A 1 A vr ' vr ' v' ' vr vr r A 1 vG Deflection angle: r '.r vr Velocity of the c.o.m. conserved Velocity modified only in direction in the c.o.m. system 4 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 Minimum energy of a n after a collision v' ( ' Ar ) We have v vG vr A 1 E v 2 A2 2 A r 1 2 E ' v' ( A 1) 2 Element H 0 D2 0.111 A 1 Emin E' E' A 1 C 0.716 U238 0.983 2 Thus Relations between variables o 1 v f (r ) o v . ' 1 v' 1 E' ( ' Ar ). ' ( A r 1) v A 1 A 1 E r f (E ) ( A 1) 2 E A2 1 r 2 A E' 2A o f (E ) o A 1 E A 1 E' 2 E' 2 E r f (o ) r 1 2 ( o o A2 1 o2 1) A o Ar 1 A2 2 Ar 1 5 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 SCATTERING LAW = probability density function (pdf) of the deflection angle Usually given in the c.o.m. system Isotropic scattering (c.o.m.) : p( r )d r 1 d r 2 In the lab system: p(r )dr p(o )do 1 A2 1 2 o2 2 p( o ) 2o 2 2A A 1 o For A=1 : 2o p( o ) | o | o 0 1 A 1 2 (cause vG small) si o 0 si o 0 Forward scattering only Slowing-down kernel (i.e. pdf of the energy of the scattered n) – isotropic case K ( E | E' )dE p(r )dr 1 if E ' E E ' K ( E | E ' ) (1 ) E ' 0 else 6 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 Mean energy loss via elastic collision E ' E E' E' with E’ (1 ) with A because 2 (1 ) E ' ( E ' E ) K ( E | E ' )dE 2 2A ( A 1) 2 A (1-)/2 1 0.5 238 0.0083 LETHARGY Eo u ln E Eo : Eréf s.t. u>0 E Eo = 10 MeV Elastic slowing-down kernel (isotropic scattering) e ( u u ') K (u | u' )du K ( E | E ' )dE du 1 with E ' E E ' u ' u u ' ln 1 1 ln q 7 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 Mean lethargy increment via elastic collision u u ' u ' ln u' 1 (u u ' ) K (u | u ' )du 1 1 ln 1 Independent of u’! As ( A 1) 2 A 1 1 ln , 2A A 1 =1 for A=1 Mean nb of collisions for a given lethargy increase: n s.t. u=n Moderator quality large + important scattering Moderating power: s Large moderation power + low absorption Moderating ratio: s/a 8 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 Moderator A n s s/a H D H 2O D 2O C U238 1 2 0 0.111 1 0.725 0.920 0.509 0.158 0.008 14 20 16 29 91 1730 1.35 0.176 0.060 0.003 71 5670 192 0.0092 12 238 0.716 0.983 u s.t. 2 MeV 1 eV a thermal 9 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 DIFFERENTIAL CROSS SECTIONS Link between differential cross section and total scattering cross section slowing-down kernel s (r , u' u) s (r , u' ) K (u | u' ) Differential cross section in lethargy and angle: s (r , u ' , ' u, ) s (r , u ' ) K (u | u ' ) 1 f ( o ) 2 Cosinus of the deflection angle: determined by the elastic collision kinematics A 1 E A 1 E' o 2 E' 2 E A 1 o (u u ' ) e 2 u 'u 2 A 1 e 2 u u ' 2 f (o ) (o o (u u' )) Deflection angle determined by the lethargy increment! 10 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 VII.3 SLOWING-DOWN EQUATION P1 APPROXIMATION Comments Objective of the n slowing down: energy spectrum of the n in the domain of the elastic collisions Input for multi-group diffusion But no spatial variation of the flux no current no diffusion! Allowance to be given – even in a simple way – to the spatial dependence One speed case: D 1 Here p( o ) 1 2A 3 tr 2 A 1 2 o2 A 1 2 2 o tr t o s with 2o with <o>0 (mainly if A1) 11 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 Steady-state Boltzmann equation in lethargy divJ (r , u, ) t (r , u ) (r , u, ) u o s (r , u ' , ' u, ) (r , u ' , ' )du' d' S (r , u, ) d d 4 (inelastic scattering accounted for in S (outside energy range)) S (r , u, ) 1 4 in (r , u ' u ) (r , u ' )du' 1 (u ) 4 4 Weak anisotropy u o f (r , u ' ) (r , u ' , ' )du' d 'Q(r , u, ) (r , u, ) 1 ( (r , u ) 3.J (r , u )) 4 0th-order momentum 1st-order momentum (S isotropic) divJ (r , u) t (r , u ) (r , u) 1 (r , u ) t (r , u ) J (r , u ) 3 u s (r , u' u) (r , u' )du' S (r , u) o u s1 (r , u ' u ) J (r , u ' )du' 0 o with s1 (r , u' u) s (r , u' , ' u, )'.d 12 For a mixture of isotopes: s (r , u' u) si (r , u' ) Ki (u | u' ) PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 i s1 (r , u' u) si (r , u' ) Ki (u | u' )oi (u u' ) i Rem: energy domain of interest: resonance absorptions Elastic collisions only Inelastic scattering: fast domain impact on the source SLOWING-DOWN DENSITY Angular slowing-down density = nb of n (/volume.t) slowed down above lethargy u in a given point and direction: q(r , u, ) u o ' u s (r , u' , ' u", )du" (r , u' , ' )du' d' Slowing-down density: q( r , u ) u o u o s (r , u' u" )du" (r , u' )du' K (u"| u' )du" s (r , u' ) (r , u' )du' u u 13 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 Slowing-down current density: q1 (r , u) u o u u o ' o u s (r , u' , ' u", )du" (r , u' , ' )du' d' d s1 (r , u' u" )du" J (r , u' )du' K (u"| u' )du" s1 (r , u' ) J (r , u' )du' u u Slowing-down density variation: u q(r , u ) s (r , u ) (r , u ) s (r , u ' u ) (r , u ' )du ' o u (interpretation?) 0th-order momentum: divJ (r , u ) ne (r , u ) (r , u ) q(r , u ) S (r , u ) u with ne (r , u) t (r , u) s (r , u) a (r , u) in (r , u) a (r , u ) domain resonance 14 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 Slowing-down current density variation u q1 (r , u ) s1 (r , u ) J (r , u ) s1 (r , u ' u ) J (r , u ' )du ' o u 1 q (r , u ) (r , u ) tr (r , u ) J (r , u ) 1 0 3 u with tr (r , u) t (r , u) s1 (r , u) t (r , u) oi si (r , u) i Slowing-down equations: summary Outside the thermal and fast domains: 1 2 3 4 q(r , u ) S (r , u ) u K (u"| u' )du" s (r , u' ) (r , u' )du' divJ (r , u ) ne (r , u ) (r , u ) q(r , u ) u o u 1 q1 (r , u ) (r , u ) tr (r , u ) J (r , u ) 0 3 u u q1 (r , u) K (u"| u' )du" s1 (r , u' ) J (r , u' )du' o u 15 INFINITE HOMOGENEOUS MEDIA PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 Without spatial dependence: t (u) (u) s (u' u) (u' )du' S (u) F (u) t (u) (u) Collision density: u Scattering probability with isotope i: o (units?) ci (u ) si (u ) t (u ) For an isotropic scattering: F (u) i 1 1 i u max(0,u qi ) e u 'u ci (u ' ) F (u' )du' S (u) with qi ln (interpretation?) 1 i Rem: F(u) and ci(u) smoother than t(u) and (u) dq (u ) S (u ) a (u ) (u ) Slowing-down density: du u Without absorption : q(u) o (u)du So for a source S (u) (u)So q(E)/So = proba not to be absorbed between Esource and E = resonance escape proba if E = upper bound of thermal E 16 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 VII.4 SLOWING DOWN IN HYDROGEN HYPOTHESES Infinite media Absorption in H neglected Slowing down considered in the resonance domain Slowing down due to heavy nuclei neglected: Elastic: minor contribution Inelastic: outside the energy range under study + low proportion of heavy nuclei FLUX SHAPE dF (u ) dS (u ) (1 c(u )) F (u ) S (u ) du du u F (u) eu 'u c(u' ) F (u' )du' S (u) o u F (u ) S (u ) e u ' u o (1 c ( u ")) du " ( F (0) 0) c(u ' ) S (u ' )du ' 17 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 One speed source S (u) So (u uo ) F (u ) e u uo a ( u ') du ' t ( u ') s (uo ) So t (uo ) for u > uo Superposition of solutions of this type for a general S Without absorption: So (u) t (u) (E) So t ( E ) E With absorption: Same behavior for (E) outside resonances (a negligible) Reduction after each resonance by a factor a ( u ') du ' e res t (u ') On the whole resonance domain, flux reduced by e u uo a (u ') du ' t (u ') e resu res a ( u ') du ' t ( u ') 18 SLOWING-DOWN DENSITY SHAPE PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 u From the definition : q(u) o u eu 'u"du" c(u' ) F (u' )du' u eu 'u c(u' ) F (u' )du' F (u ) S (u ) o One speed source (uo) and u > uo q(u) e u uo (1c (u ')) du ' c(uo )So Resonance escape proba in u: u a (u ' ) q(u ) p(u ) exp du' q(uo ) uo t (u ' ) 19 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 VII.5 OTHER MODERATORS Reminder: homogeneous media F (u) i 1 1 i u max(0,u qi ) eu 'u ci (u ' ) F (u' )du' S (u) PLACZEK FUNCTION P(u) = collision density F(u) iff One material No absorption One speed source P(u) K (u u' ) P(u' )du' (u) o with K (u ) 1 u e H (u ) H (q u ) 1 20 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 1 p 1 Laplace K ( p) (1 )(1 p) P ( p) K ( p) P ( p) 1 1 1 K ( p) K 2 ( p) ... P ( p) 1 K ( p) Inverting term by term, effect of an increasing nb of collisions u eu 'u P(u ' )du' ? Solution of P(u ) (u ) max(0 ,u q ) 1 By intervals of width q At the origin: P(u ) (u) u exp( ) 1 1st interval 0 < u < q : P(u) 1 Discontinuity in q : 1 1 u exp( ) 1 1 nd (1 (u q)) 2 interval q < u < 2q : P(u ) 1 1 21 Tauber’s theorem P lim pP ( p ) p 0 p 1 p 0 1 K ( p ) lim (1-)P(u) PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 Asymptotic behavior Oscillations in the neighborhood of the origin =Placzek oscillations u/q 22 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 SYNTHETIC SLOWING-DOWN KERNELS ordinary diff. eq. for H diff. eq. with delay else Approximations to simplify this diff. eq. Integral slowing-down equation Wigner approximation Asymptotic behavior of F(u) for an absorbing moderator, with c(u) cst, for a one speed S: 1c 1 u Fas (u) e c Approx. for a slow variation of c(u): u t (u ) o Fas (u ) e s (u ) (c1) a ( u ') du ' t ( u ') 23 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 Slowing-down density: q(u) u o u u u u q u q K (u"| u' )du" c(u' ) F (u' )du' eu 'u " u du" c(u ' ) F (u ' )du' 1 eu ' u c(u ' ) F (u ' )du' 1 u ' q Asymptotic zone (Wigner): qas (u ) e u qas (u) e 1 u q u 'u u o a (u ') du ' t (u ') (u>q) du' c(u ) Fas (u ) p(u) Resonance escape proba in u 24 Justification of the approximation PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 Mean nb of collisions to cross ui where c(ui) cst: ui/ Proba to cross without absorption n consecutive intervals ui/n : u (u ) u u (1 i . a i n t (ui ) p(u) i ) n (1 i n 1 (1 c(ui )))n n exp( i (1 c(ui ))) 1 u exp( (1 c(ui ))) exp (1 c(u' ))du' o ui Variation in the approximation Outside the source domain: a (u ) dq as (u ) a (u ) a (u ) (u ) qas (u ) c(u ) Fas (u ) du s (u ) s (u ) Age approximation (see below) 1c 1 c u Rem: compatible with Fas (u) e for any c c 25 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 Greuling-Goertzel approximation d c(u ' ) F (u ' ) c(u ) F (u ) (u 'u ) (c(u ) F (u )) we consider du d q ( u ) c ( u ) F ( u ) (c(u ) F (u )) In the asymptotic zone du 1 dq (u ) with u 2 c ( u ) F ( u ) 2 du dq (u ) Yet a (u ) (u ) du Thus (u ) q(u ) s (u ) a (u ) a (u )q(u ) dq(u ) du s (u ) a (u ) Resonance escape proba p(u) e u o a (u ') du ' s (u ')a (u ') Rem: Wigner if Age if 0 26 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 Generalization: synthetic kernels Objective: replace the integral slowing-down eq. F (u) K (u u' )c(u' ) F (u' )du' S (u) o by an ordinary differential eq. (i.e. without delay) ~ Synthetic kernel K (u u' ) close to the initial kernel K (u u' ) and s.t. approximated solution close enough to F(u) Close? Momentums conservation: ~ k M k K (u)u du K (u)u k du o o k m Choice of the synthetic kernel? Solution of d Lm (u) k ~ du k 0 Lm (u)K (u u' ) Dn (u) (u u' ) with k n d Dn (u) k du k 0 approximated diff.eq. for the slowing-down density: ~ ~ Lm (u)(F (u) S (u)) Dn (u)c(u)F (u) 27 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 Parameters of the differential operators Lm(u) and Dn(u)? Conservation of m+n+1 momentums 1st-order synthetic kernels: m=1, n=0 Wigner m=0, n=1 age m=1, n=1 Greuling – Goertzel 28 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 VII.6 SPATIAL DEPENDENCE Slowing down in finite media FERMI’S AGE THEORY Use of the P1 equations with • the age approximation: q(r , u) (u)s (u) (r , u) q1 ( r , u ) • neglected in the current equation u 1 J (r , u ) ( r , u ) D( r , u ) ( r , u ) 3tr (r , u ) and div ( D(r , u ) (r , u )) a (r , u ) (r , u ) q(r , u ) S (r , u ) u homogeneous zone, beyond the sources: (r , u ) D(u ) q(r , u ) q(r , u ) a q(r , u ) 0 (u ) s (u ) (u ) s (u ) u 29 u PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 Let (u) o D(u ' ) du' , with : n age [cm2] !! (u ' ) s (u' ) 1 q(r , ) q(r , ) 2 q ( r , ) 0 L ( ) Let q(r , ) p( )q~(r , ) with the resonance escape proba p( ) e Fermi’s equation u ( ) o a ( u ') du ' s ( u ') e o 1 d ' L2 ( ') q~ (r , ) : slowing-down density without absorption q~ (r , ) ~ q ( r , ) Equivalent to a time-dependent diffusion equation! 30 Relation lethargy – time ? PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 Heavy nuclei mean lethargy increment low low dispersion of the n lethargies same moderation If slowing down identical for all n, u = f(slowing-down time) With all n with the same lethargy, the diffusion equation at time t writes (for n emitted at t=0 with u=0): 1 (r , t ) div ( D(r , t ) (r , t )) a (r , t ) (r , t ) 0 v t Variation of u / u.t.: du s vdt Fermi’s equation Approximation validity Moderators heavy enough graphite in practice 31 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 Examples of slowing-down kernels Planar one speed source (Eo) IC: q~(r ,0) ( x xo ) e ~ q ( r , ) | x xo |2 4 4 Point one speed source (Eo) ~ IC: q (r ,0) (r ro ) |r ro |2 4 e ~ q ( r , ) (4 )3 / 2 Mean square distance to the source: r2 o r 2 q~(r , )4r 2 dr o q~(r , )4r 2 dr 6 Age = measure of the diffusion during the moderation 32 PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 Consistent age theory Same treatment for q1 (r , u) as for q(r , u ) q1 (r , u) 1 (u)s (u) J (r , u) with 1 u o u K1 (u"u' )du" du' K1 (u) K (u)o (u) A 1 2 A 1 2 o (u ) e e 2 2 u u 33