1. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 1 CH.II: NEUTRON TRANSPORT INTRODUCTORY CONCEPTS ASSUMPTIONS NEUTRON DENSITY, FLUX, CURRENT REACTION RATE FLUENCE, POWER, BURNUP TRANSPORT EQUATION NEUTRON BALANCE BOLTZMANN EQUATION CONTINUITY AND BOUNDARY CONDITIONS INTEGRAL FORMS FORMAL SOLUTION USING NEUMANN SERIES

2. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 2 II.1 INTRODUCTORY CONCEPTS ASSUMPTIONS 1.Interactions between n – matter: quantum problem But (n) << characteristic dimensions of the reactor  E 2.Density of thermal n: ~ 10 9 n/cm 3 Atomic density of solids: ~ 10 22 atoms/cm 3  Interactions n – n negligible  Linear equation for the neutron balance 3.Statistical treatment of the n, but small fluctuations about the average value of their flux  n: classical particles, not interacting with each other, whose average value of their probability density of presence is accounted for

3. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 NEUTRON DENSITY, FLUX, CURRENT Variables Position: 3 Speed: 3or kinetic energy + direction (Time: 1t) Angular neutron density : nb of n in about with a speed in [v,v+dv] and a direction in about Neutron density Angular neutron density whatever the direction 3 (similar definition with variables (r,E,  ) or (r,v)) (dimensions of N in both cases?)

4. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 4 Angular flux s.t. Total neutron flux Integrated flux (Angular) current density  Nb of n flowing through a surface  / u.t. (net current) Isotropic distribution? [dim ?] Net current = 0 , hence flux spatially c st ? Wrong!!  A reactor is anisotropic  But weak anisotropy often ok (1 st order) [dim ?]

5. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 5 REACTION RATE Nb of interactions / (volume.time): R Beam of incident n on a (sufficiently) thin target (  internal nuclei not hidden):  R = N.  (r,t).  (r) =  (r).  (r,t) Or: interaction frequency = v  [s -1 ] n density = n(r,t)[m -3 ]  R = n(r,t).v  (r) =  (r).  (r,t) General case: cross sections dependent on E (  v) Rem:  = f(relative v between target nucleus and n) while  = f(absolute v of the n)  implicit assumption (for the moment): heavy nuclei immobile (see chap. VIII to release this assumption) R = Nb nuclei cm 3 Nb n cm 2.s Cross sectional area of a nucleus (cm 2 ) xx

6. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 6 Rem: differential cross sections Scattering  speed after a collision?  Conditional probability that 1 n with speedundergoing 1 collision at leaves it with a speed in [v’, v’+dv’] and a direction in about with  Scattering kernel s.t. Isotropic case: [dim ?] Why?

7. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 7 FLUENCE, POWER, BURNUP Fluence  Characteristics of the irradiation rate ([n.cm -2 ] or [n.kbarn -1 ]) Power Linked to the nb of fission reactions Burnup Thermal energy extracted from one ton of heavy nuclei in fresh fuel  Fluence x x energy per fission

8. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 8 II.2 TRANSPORT EQUATION NEUTRON BALANCE Variation of the nb of n (/unit speed) in volume V, in dv about v, in about Sources Losses due to all interactions Losses through the boundary  Gauss theorem V  (n produced in about, dv about v, about ) (n lost in about, dv about v, about ) dS

9. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 9 Rem: general form of a conservation equation BOLTZMANN EQUATION (transport) Without delayed n Sources? Steady-state form Scattering External source Fission Total nb/(vol. x time) of n due to all fission at r Fraction in dv about v, d  about 

10. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 10 Compact notation with (destruction-scattering operator) and (production operator) Non-stationary form

11. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 11 With delayed n Concentration C i of the precursors of group i: with i = (ln 2) / T i Def: production operator for the delayed n of group i, i = 1…6 : Total production operator: Let prompt n Fraction/(vol. x time) of n due to all fissions at r in group i Radioactive decay Fraction in dv about v, d  about  Production of delayed n of group i / (vol. x time) (precursors assumed to be a direct product of a fission)

12. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 12 System of equations for the transport problem with delayed n Stationary regime Reduction to 1 equation: Production operator: equivalent to having J  J o (prompt n) iff  Formalism equivalent, with or without delayed n, with a modified fission spectrum (Why in stationary regime?)

13. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 13 CONTINUITY AND BOUNDARY CONDITIONS Nuclear reactors: juxtaposition of uniform media (   indep. of the position)  How to combine solutions of in the  media? Let  : discontinuity border (without superficial source) Integration on a distance [- ,  ] about in the direction continuity on  Boundary condition (convex reactor surrounded by an  vacuum):

14. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 14 INTEGRAL FORMS If s = distance covered in the direction of the n: Lagrange’s variation of constants: Let (interpretation ?) : optical thickness (or distance) [1]

15. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 15 Yet  If both scattering and independent source are isotropic  After integration to obtain the total flux: Rem: S fct of  !!

16. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 16 Explicit form of the integral equation for the angular flux We have and Thus (interpretation ?)

17. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 17 Transition kernel  Transport process: proba distribution of the ingoing coordinates in the next collision, given the outgoing coordinates from the previous one Collision kernel  Impact: entry in 1 collision  exit Compact notation :  Captures not considered  Fissions :  1  = 1 for an infinite reactor (based on the negative exponential law)

18. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 18 Collision densities Ingoing density:= expected nb of n entering/u.t. in a collision with coordinates in dP about P Outgoing density:= expected nb of n leaving/u.t a collision with coordinates in dP about P Evolution equations Interpretation ?

19. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 19 Rem:  equ. of  (P) = (equ. of  (P)) x  t (P)  Possible interpretation of n transport as a shock-by-shock process FORMAL SOLUTION USING NEUMANN SERIES Let   j (P): ingoing collision density in the j th collision  : solution of the transport equation  Not realistic: infinite summation…  Basis for solution algorithms

20. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 20 CH.II: NEUTRON TRANSPORT INTRODUCTORY CONCEPTS ASSUMPTIONS NEUTRON DENSITY, FLUX, CURRENT REACTION RATE FLUENCE, POWER, BURNUP TRANSPORT EQUATION NEUTRON BALANCE BOLTZMANN EQUATION CONTINUITY AND BOUNDARY CONDITIONS INTEGRAL FORMS FORMAL SOLUTION USING NEUMANN SERIES  