Critical Assembly Computer Laboratory
Overview:
In this laboratory, you will use the PENTRAN (Parallel Environment Neutralparticle TRANsport) on
your laptop; it is a finite difference code to solve the Boltzmann Transport Equation (BTE), to be
solved for accurate computation of critical assemblies and radiation shielding problems.
You will use
PENTRAN to solve for the neutron multiplication factor (
k
) for various gap distances separating two
large cubes (LxWxH=15 cm x 15 cm x 12 cm) of 93% enriched uranium235.
Any arrangement of
fissile materials like this is known as a “critical assembly.”
Part I:
PreLab
The BTE tracks neutrons traveling through a system in a discrete number of vector directions, each
located at specific points in 3dimensionsional space, and in a specific energy “bin” or “group”
g
.
The 3D
xyz
BTE, discretized in angle (direction) and energy group, is (using spherical harmonics
with the Legendre Addition Theorem for the scattering term):
=
+
∂
∂
+
∂
∂
+
∂
∂
)
,
,
,
,
(
)
,
,
(
)
,
,
,
,
(
)
(
ϕ
μ
ψ
σ
ξ
η
z
y
x
z
y
x
z
y
x
z
y
x
g
g
g
∑∑
∑
=
=
=
→
⋅
+

+
+
G
g
L
l
l
k
k
l
l
g
l
l
g
g
s
P
k
l
k
l
z
y
x
P
z
y
x
l
1
'
0
1
0
0
,
'
,
'
,
0
0
)
(
)!
(
)!
(
2
)
,
,
(
)
(
){
,
,
(
)
1
2
(
φ
∑
=
+
+
G
g
g
g
f
g
k
l
g
S
k
l
g
C
z
y
x
z
y
x
k
k
z
y
x
k
z
y
x
1
'
0
,
'
'
0
,
'
0
,
'
)
,
,
(
)
,
,
(
)]}
sin(
)
,
,
(
)
cos(
)
,
,
(
[
0
0
νσ
χ
Although this is a complicated equation, like all differential equations, it is a
balance equation
.
From left to right, the terms in the BTE are, for a neutron in a specific direction and energy group:
(Leakage of neutrons out of the system from this direction)
+ (Collisions of neutrons from this direction that result in scatter or absorption)
=
(Scatter of neutrons into this direction, energy from all other energy groups and directions)
+ (Production of fission neutrons into this direction from all other groups and directions)/
k
The neutrons that emerge from fission are “
fast
” neutrons (which means in the MeV energy range),
and as they collide and scatter with other nuclei (especially lighter elements), they can lose energy, all
the way down into the 0.01 eV range.
As they lose energy, the probability they will cause a fission in
fissile materials greatly increases.
In 1945, the scientists at Los Alamos did not have very accurate
numbers for the interaction probabilities (
’s) known as “crosssections” in the BTE (this is
understandable, since one must consider a range of eight orders of magnitude in energy!).
Since then,
with precise measurements and computer models, we have a better idea of these socalled “multi
group crosssections.”
We will be modeling our criticality problems with a 16energy group Los
Alamos Hansen Roach cross section set that has been collapsed down to just 2 energy groups: