1. CH.IV : CRITICALITY CALCULATIONS IN DIFFUSION THEORY
ONE-SPEED DIFFUSION
MODERATION KERNELS
REFLECTORS
INTRODUCTION
REFLECTOR SAVINGS
TWO-GROUP MODEL

2.  A time-independent  can be sustained in the reactor with no Q
IV.1 CRITICALITY
Objective
solutions of the diffusion eq. in a finite homogeneous
media exist without external sources
1st study case: bare homogeneous reactor (i.e. without reflector)
ONE-SPEED DIFFUSION
With fission !!
Helmholtz equation
with
and BC at the extrapolated boundary:
  : solution of the corresponding eigenvalue problem
countable set of eigenvalues:
criticality
 A time-independent  can be sustained in the reactor with no Q

3. + associated eigenfunctions: orthogonal basis
A unique solution positive everywhere  fundamental mode
Flux !
Eigenvalue of the fundamental – two ways to express it:
= geometric buckling
= f(reactor geometry)
= material buckling
= f(materials)
Criticality:
Core displaying a given composition (Bm cst): determination of the size (Bg variable) making the reactor critical
Core displaying a given geometry (Bg cst): determination of the required enrichment (Bm)

5. Time-dependent problem
Diffusion operator:
 Spectrum of real eigenvalues:
s.t. with
o = maxi i associated to : min eigenvalue of (-)  o associated to o: positive all over the reactor volume
Time-dependent diffusion:
Eigenfunctions i: orthogonal basis 
o < 0 : subcritical state
o > 0 : supercritical state
o = 0 : critical state with J -K

6. and criticality for keff = 1
Unique possible solution of the criticality problem whatever the IC:
Criticality and multiplication factor
keff : production / destruction ratio
Close to criticality:
o = fundamental eigenfunction associated to the eigenvalue keff of:
 media:
Finite media:
Improvement: with
and criticality for keff = 1

7. Independent sources
Eigenfunctions i : orthonormal basis
Subcritical case with sources: possible steady-state solution
Weak dependence on the expression of Q, mainly if o(<0)  0
Subcritical reactor: amplifier of the fundamental mode of Q
 Same flux obtainable with a slightly subcritical reactor + source as with a critical reactor without source

8. Solution in an  media: use of Fourier transform
MODERATION KERNELS
Definitions
= moderation kernel: proba density function that 1 n due to a fission in is slowed down below energy E in
= moderation density: nb of n (/unit vol.time) slowed down below E in (see chap.VII)
with
 media: translation invariance 
Finite media: no invariance  approximation
Solution in an  media: use of Fourier transform
Objective: improve the treatment of the
dependence on E w.r.t. one-speed diffusion

9. Solution in finite media
Inverting the previous expression:
solution of
Solution in finite media
Additional condition: B2  {eigenvalues} of (-) with BC on the extrapolated boundary 
Criticality condition:
with solution of
 : fast non-leakage proba

10. Examples of moderation kernels
Reminder: convolution product in r  product of Fourier transforms in B
Two-group diffusion
Fast group: 
Criticality eq.:
G-group diffusion
Age-diffusion (see Chap.VII)  
 Criticality eq.:
(E) = age of n at en. E emitted at the fission en.
 = age of thermal n emitted at the fission en.

11. IV.2 REFLECTORS INTRODUCTION No bare reactor Thermal reactors
backscatters n into the core
Slows down fast n (composition similar to the moderator)
 Reduction of the quantity of fissile material necessary to reach criticality  reflector savings
Fast reactors
n backscattered into the core? Degraded spectrum in E
 Fertile blanket (U238) but  leakage from neutronics standpoint
 Not considered here

12. One-speed diffusion model
REFLECTOR SAVINGS
One-speed diffusion model
In the core:
 with
In the reflector: 
Solution of the diffusion eq. in each of the m zones  solution depending on 2.m constants to be determined
Use of continuity relations, boundary conditions, symmetry constraints… to obtain 2.m constraints on these constants
Homogeneous system of algebraic equations: non-trivial solution iff the determinant vanishes
Criticality condition

13. Solution in planar geometry
Consider a core of thickness 2a and reflector of thickness b (extrapolated limit)
Problem symmetry 
Flux continuity + BC:
Current continuity:
 criticality eq.
Q: A = ?

14. Criticality reached for a thickness 2a satisfying this condition
For a bare reactor:
 Reflector savings:
In the criticality condition:
As Bc << 1 :
If same material for both reflector and moderator, with a D little affected by the proportion of fuel  D  DR
Criticality: possible calculation with bare reactor accounting for 

15. TWO-GROUP MODEL Core Reflector Planar geometry: solutions s.t. ?
Solution iff determinant = 0
 2nd-degree eq. in B2
 (one positive and one negative roots)
For each root:

16. Solution in the core for [-a, a]:
Solution in the reflector for a  x  a+b:
4 constants + 4 continuity equations (flux and current in each group)
Homogeneous linear system
Annulation of the determinant to obtain a solution
Criticality condition
Q: the flux is then given
to a constant. Why?

17. fast flux
thermal flux
core
reflector

18. CH.IV : CRITICALITY CALCULATIONS IN DIFFUSION THEORY
ONE-SPEED DIFFUSION
MODERATION KERNELS
REFLECTORS
INTRODUCTION
REFLECTOR SAVINGS
TWO-GROUP MODEL  