1. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 1 CH.VII: NEUTRON SLOWING DOWN INTRODUCTION SLOWING DOWN VIA ELASTIC SCATTERING KINEMATICS SCATTERING LAW LETHARGY DIFFERENTIAL CROSS SECTIONS SLOWING-DOWN EQUATION P 1 APPROXIMATION SLOWING-DOWN DENSITY INFINITE HOMOGENEOUS MEDIA SLOWING DOWN IN HYDROGEN HYPOTHESES FLUX SHAPE SLOWING-DOWN DENSITY SHAPE

2. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 2 OTHER MODERATORS PLACZEK FUNCTION SPATIAL DEPENDENCE FERMI’S AGE THEORY SLOWING DOWN IN HYDROGEN APPENDIX: SYNTHETIC SLOWING-DOWN KERNELS IN INFINITE HOMOGENEOUS MEDIA

3. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 3 VII.1 INTRODUCTION Decrease of the n energy from E fission to E th due to possibly both elastic and inelastic collisions Inelastic collisions: E of the incident n > 1 st excitation level of the nucleus 1 st excited state for light nuclei: 1 MeV 1 st 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

4. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 G KINEMATICS Absolute coordinates of nc.o.m. system  Velocity of the c.o.m. conserved  Velocity modified only in direction in the c.o.m. system G Before collisionAfter collision Deflection angle: 4 VII.2 SLOWING DOWN VIA ELASTIC SCATTERING Before collisionAfter collision E’E Deflection angle: n A

5. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 5 Minimum energy of a n after a collision We have Thus Relations between variables Element  H0 D2D2 0.111 C0.716 U 238 0.983

6. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 6 SCATTERING LAW = probability density function (pdf) of the deflection angle Usually given in the c.o.m. system Isotropic scattering (c.o.m.) : In the lab system: (cause v G small) For A=1 :  Forward scattering only Slowing-down kernel (i.e. pdf of the energy of the scattered n) – isotropic case

7. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 7 Mean energy loss via elastic collision  with E’  with A because LETHARGY E o : E réf s.t. u>0  E  E o = 10 MeV Elastic slowing-down kernel (isotropic scattering) with A (1-  )/2 10.5 2380.0083

8. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 8 Mean lethargy increment via elastic collision Independent of u’! As,  =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

9. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 9  u s.t. 2 MeV  1 eV  a thermal ModeratorA  n  s  s /  a H D H 2 O D 2 O C U 238 1 2 12 238 0 0.111 0.716 0.983 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

10. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 10 DIFFERENTIAL CROSS SECTIONS Link between differential cross section and total scattering cross section  slowing-down kernel Differential cross section in lethargy and angle: Cosinus of the deflection angle: determined by the elastic collision kinematics  Deflection angle determined by the lethargy increment!

11. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 11 VII.3 SLOWING-DOWN EQUATION P 1 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:with Here with  0 (mainly if A  1)

12. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 12 Steady-state Boltzmann equation in lethargy (inelastic scattering accounted for in S (outside energy range)) Weak anisotropy  0 th -order momentum 1 st -order momentum (S isotropic)  dd dd with

13. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 13 For a mixture of isotopes:  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: Slowing-down density:

14. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 14 Slowing-down current density: Slowing-down density variation: (interpretation?)  0 th -order momentum: with resonance domain

15. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 15 Slowing-down current density variation  with Slowing-down equations: summary Outside the thermal and fast domains: 1 2 3 4

16. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 16 INFINITE HOMOGENEOUS MEDIA Without spatial dependence: Collision density: Scattering probability with isotope i:  For an isotropic scattering: with Rem: F(u) and c i (u) smoother than  t (u) and  (u) Slowing-down density: Without absorption : for a source  q(E)/S o = proba not to be absorbed between E source and E = resonance escape proba if E = upper bound of thermal E (interpretation?) (units?)

17. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 17 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

18. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 18 One speed source  for u > u o  Superposition of solutions of this type for a general S Without absorption: With absorption:  Same behavior for  (E) outside resonances (  a negligible)  Reduction after each resonance by a factor  On the whole resonance domain, flux reduced by

19. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 19 SLOWING-DOWN DENSITY SHAPE From the definition : One speed source (u o ) and u > u o   Resonance escape proba in u:

20. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 20 VII.5 OTHER MODERATORS Reminder:  homogeneous media PLACZEK FUNCTION P(u) = collision density F(u) iff  One material  No absorption  One speed source with

21. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 21 Laplace   Inverting term by term, effect of an increasing nb of collisions Solution of?  By intervals of width q  At the origin:  1 st interval 0 < u < q :  Discontinuity in q :  2 nd interval q < u < 2q :

22. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 22 Asymptotic behavior  Tauber’s theorem Oscillations in the neighborhood of the origin =Placzek oscillations Appendix: Synthetic slowing- down kernels in infinite homogeneous media (1-  )P(u) u/q

23. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 23 VII.6 SPATIAL DEPENDENCE  Slowing down in finite media FERMI’S AGE THEORY Use of the P 1 equations with the age approximation: neglected in the current equation  homogeneous zone, beyond the sources: and

24. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 24 Let, with  : n age [cm 2 ] !! Let with the resonance escape proba  : slowing-down density without absorption  Equivalent to a time-dependent diffusion equation! Fermi’s equation

25. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 25 Relation lethargy – time ? 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): Variation of u / u.t.:  Fermi’s equation Approximation validity Moderators heavy enough  graphite in practice

26. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 26 Examples of slowing-down kernels Planar one speed source (E o ) IC:  Point one speed source (E o ) IC:  Mean square distance to the source:  Age = measure of the diffusion during the moderation

27. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 27 Consistent age theory Same treatment for as for with

28. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 28 APPENDIX: SYNTHETIC SLOWING-DOWN KERNELS IN INFINITE HOMOGENEOUS MEDIA Integral slowing-down equation  ordinary diff. eq. for H  diff. eq. with delay else  Approximations to simplify this diff. eq. Wigner approximation Asymptotic behavior of F(u) for an absorbing moderator, with c(u) c st, for a one speed S:  Approx. for a slow variation of c(u): (c1)(c1)

29. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 29 Slowing-down density: Asymptotic zone (Wigner):  Resonance escape proba in u (u>q) 

30. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 30 Justification of the approximation Mean nb of collisions to cross  u i where c(u i )  c st :  u i /   Proba to cross without absorption n consecutive intervals  u i /n : Variation in the approximation Outside the source domain:  Age approximation (see below) Rem: compatible withfor any c

31. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 31 Greuling-Goertzel approximation we consider In the asymptotic zone  with Yet Thus Resonance escape proba Rem: Wigner if  Age if  0

32. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 32 Generalization: synthetic kernels Objective: replace the integral slowing-down eq. by an ordinary differential eq. (i.e. without delay)  Synthetic kernel close to the initial kernel and s.t. approximated solution close enough to F(u)  Close? Momentums conservation: Choice of the synthetic kernel? Solution of  approximated diff.eq. for the slowing-down density: with

33. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 33 Parameters of the differential operators L m (u) and D n (u)?  Conservation of m+n+1 momentums 1 st -order synthetic kernels:  m=1, n=0  Wigner  m=0, n=1  age  m=1, n=1  Greuling – Goertzel

34. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 34 CH.VII: NEUTRON SLOWING DOWN INTRODUCTION SLOWING DOWN VIA ELASTIC SCATTERING KINEMATICS SCATTERING LAW LETHARGY DIFFERENTIAL CROSS SECTIONS SLOWING-DOWN EQUATION P 1 APPROXIMATION SLOWING-DOWN DENSITY INFINITE HOMOGENEOUS MEDIA SLOWING DOWN IN HYDROGEN HYPOTHESES FLUX SHAPE SLOWING-DOWN DENSITY SHAPE    

35. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 35 OTHER MODERATORS PLACZEK FUNCTION SPATIAL DEPENDENCE FERMI’S AGE THEORY SLOWING DOWN IN HYDROGEN APPENDIX: SYNTHETIC SLOWING-DOWN KERNELS IN INFINITE HOMOGENEOUS MEDIA   