CH.II: NEUTRON TRANSPORT

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. II.1 INTRODUCTORY CONCEPTS ASSUMPTIONS • Interactions between n – matter: quantum problem But (n) << characteristic dimensions of the reactor E • Density of thermal n: ~ 109 n/cm3 Atomic density of solids: ~ 1022 atoms/cm3 • Interactions n – n negligible • Linear equation for the neutron balance • 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. NEUTRON DENSITY, FLUX, CURRENT Variables Position: 3 Speed: 3 or kinetic energy + direction (Time: 1 t) 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 (same definition with variables (r,E,)) (dimensions of N in both cases?)

4. [dim ?] 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 cst? Wrong!! • A reactor is anisotropic • But weak anisotropy (1st order)

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 vbetween target nucleus and n) while  = f(absolute vof the n)  implicit assumption(for the moment): heavy nuclei immobile (see chap. VIII to release this assumption) R = Nb nuclei cm3 Nb n cm2.s Cross sectional area of a nucleus (cm2) x x

6. Rem: differential cross sections Scattering  speed after a collision? • Conditional probability that 1 n with speed undergoing 1 collision at leaves it with a speed in [v’, v’+dv’] and a direction in about with • Scattering kernel s.t. Isotropic case: Why? [dim ?]

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 <fission cross section> x energy per fission

8. II.2 TRANSPORT EQUATION dS  V 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  Gausstheorem (n produced in about , dv about v, about ) (n lost in about , dv about v, about )

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

10. Compact notation with (destruction-scattering operator) and (production operator) Non-stationary form

11. Fraction/(vol. x time) of n due to all fissions at r in group i With delayed n Concentration Ci of the precursors of group i: with i = (ln 2) / Ti Def: production operator for the delayed n of group i, i = 1…6: Total production operator: Let Radioactive decay Production of delayed n of group i / (vol. x time) prompt n Fraction in dv about v, d about 

12. System of equations for the transport problem with delayed n Stationary regime Reduction to 1 equation: Production operator: equivalent to having J  Jo (prompt n) iff  Formalism equivalent with or without delayed n

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 (convexreactor surrounded by an  vacuum):

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. Yet  If both scattering and independent source are isotropic • After integration to obtain the total flux: Rem: S fct of  !!

16. Explicit form of the integral equation for the angular flux We have and Thus (interpretation ?)

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 : (based on the negative exponential law) • = 1for an infinite reactor • Captures not considered • Fissions :   1

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. 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 jth collision  : solution of the transport equation • Not realistic: infinite summation… • Basis for solution algorithms