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

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

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 (similar definition with variables (r,E,) or (r,v)) (dimensions of N in both cases?)

s.t. Angular flux Total neutron flux Integrated flux (Angular) current density  Nb of n flowing through a surface  / u.t. (net current) Isotropic distribution? [dim ?] [dim ?] Net current = 0 , hence flux spatially cst? Wrong!! A reactor is anisotropic But weak anisotropy often ok (1st order)

Nb of interactions / (volume.time): R 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 cm3 Nb n cm2.s Cross sectional area of a nucleus (cm2) x x

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 ?]

FLUENCE, POWER, BURNUP Fluence Power Burnup 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 (expressed in GWd/tHM (gigawatt-days/metric ton of heavy metal))  Fluence x <fission cross section> x energy per fission

II.2 TRANSPORT EQUATION NEUTRON BALANCE Variation of the nb of n (/unit speed) in volume V, in dv about v, in about dS  V Sources Losses due to all interactions Losses through the boundary  Gauss theorem (n produced in about , dv about v, about ) (n lost in about , dv about v, about )

BOLTZMANN EQUATION (transport) Without delayed n 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

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

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 Fraction/(vol. x time) of n due to all fissions at r in group i (precursors assumed to be a direct product of a fission) Radioactive decay Production of delayed n of group i / (vol. x time) prompt n Fraction in dv about v, d about 

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, with a modified fission spectrum (Why in stationary regime?)

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):

INTEGRAL FORMS If s = distance covered in the direction of the n: Lagrange’s variation of constants: Let (interpretation ?) : optical thickness (or distance) [1]

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

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

 Impact: entry in 1 collision  exit 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) = 1 for an infinite reactor Captures not considered Fissions :   1

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 ?

FORMAL SOLUTION USING NEUMANN SERIES 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

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 