 diffusion of neutrons through matter
 large number of random neutron
interactions in matter
 calculate the angular, energy and
spatial variation of the neutron
distribution
(x,y,z,E,)
 study and design of nuclear reactors
 core and fuel performance
 beam and irradiation facilities
 shielding and safety
 medical physics eg. BNCT
 scattering
v1 , E1 , 1
vo, Eo, o
elastic
v1 , E1 , 1
vo, Eo, o
inelastic
vo, Eo, o
absorption eg. (n,)
vi, Ei, i
vo, Eo, o
fission
n
Flux is the number of neutrons crossing
the surface (perpendicular to the
direction of motion) per area per time
d
Flux is the total number of neutrons
passing through a surface, that is
perpendicular to each neutron direction,
per area per time. Direction doesn’t
matter.
 What is neutron transport
 The neutron transport equation
 Solving the neutron transport equation
Deterministic methods
 PL approximation
 Diffusion equation
 Discrete ordinates
Monte Carlo
 Typical reactor calculation
 Work on RRR
dz
dy
x
x + dx
neutrons going out – neutrons going in
 x ((x+dx) - (x)) dy dz
+ y ((y+dy) - (y)) dx dz
+ z ((z+dz) - (z)) dy dx
 dV


no sources or
collisions)
  sources - losses (collisions)
 Q + (E’E,’) d’dE’ - 

 (E)  (E’) f(E’) d’dE’
k
4
+ (E’E,’) d’dE’ - 

 analytic solution only for simple
geometries and even then with
approximations, such as isotropic
scattering
 provide insight to real reactors that
are similar to calculated geometries
 allow comparison of results from
other approximate techniques
 expand the angular flux in terms of
the spherical harmonics
2l  1 l
( x, y, z, E , )  
  lm ( x, y, z, E )Ylm ()
l  0 4 m   l

2l  1
s  
 sl Pl (cos )
l  0 4

 we can integrate equations over all
angles and eliminate  from equations
 for 1D cases can calculate to high
order
 for 2D and 3D usually take only the
first 2 terms ie. l = 1. This is the P1
approximation.
 starting from the P1 equations and
assuming a << we obtain
-D (E)  (E’) f(E’)dE’
k

+ (E’E) dE’ - 
J = -D Fick’s law
1
D
3(  0  s )
 good approximation for slowly varying
flux
 poor near boundaries of materials
within a few mfp.
 solve the transport equation for a set
of discrete directions m
 these directions are weighted
according to the solid angle subtended
wm = m/4
( x, y, z, E )   wm m ( x, y, z, E )
m
 not limited by large flux gradients
 can bias the direction angles towards
a direction of high importance
 formulation allows efficient coding
for numerical solution using computers


2
1
number of neutrons crossing each
surface (per time) is
A

for 1
A cos 











for


 simplify the energy representation by
using group cross-sections
 a group is an energy interval Ei  E  Ei+1
 average the point cross-sections using
some estimated neutron spectrum
 usually this is 
a fission emission spectrum
+ a 1/E spectrum
+ a Maxwellian thermal spectrum

Ei 1
Ei  ( E )( E )dE
g 
Ei 1
Ei  ( E )dE
 need to divide the space co-ordinates
into a set of discrete values
 this is the mesh
material 1
material 2
material 3
 we assume average values for  and 
within any mesh interval
 we approximate derivatives with
differences
d ( x  x)  ( x)

dx
x
 impose continuity of flux and current
across common boundaries between
mesh intervals
 obtain a set of linear equations relating
(x) and the (xi) in surrounding mesh
intervals
 solve using various methods
 inner iterations – the fission and
scattering sources are kept constant
and the group fluxes are solved
 outer iterations – the fission and
scattering sources are recalculated
 start calculation with a flat flux and
then calculate fission source
 cross-section library must be
condensed. Anything from 100’s
groups to 20-50 groups.
 too much detail in the core
fuel meat, cladding and coolant channel
 cell calculation, represent the fuel
meat, cladding and coolant channel
cladding
fuel
coolant
 perform in 1D using reflective
boundary conditions ie. infinite lattice
Use discrete ordinates
 homogenise and condense crosssections using the flux solution
  i  iVi
 I  iI
  iVi
iI
 perform a supercell calculation, whole
fuel element
 homogenise the supercell material and
perform a whole reactor calculation.
Diffusion or discrete ordinates.
 do not solve any transport equations
 simulate the neutron tracks through
the system of interest (reactor)
 there are no averages over energy,
space or time. All aspects of the
problem are simulated
 events happen at random
 sample next interaction position
Probability that a neutron travels a
distance x without interaction is
P = e-x
If P is a uniformly sampled random
number then the distance is given by
x = ln P / 
 sample interaction from all possible
interactions according to their relative
cross-sections


 depending on interaction sample the
corresponding neutron energy and
angular distribution etc.
 sample many events to accumulate
sufficient statistics to calculate the
required results
 Monte Carlo can be slow especially
reactor calculations. Most of the time
is spent calculating the source.
 usually no simple way to calculate the
flux distribution
 variance reduction techniques exist
shield
core
reflector