Reflecting Collimator Design
Introduction
The first remark that should be made concerning the design of the collimator for this
project is that the device we
seek to construct
resembles a reflector much more than a
collimator, since the principle aim of th
is device will be to enhance the flux at the target
rather than decreasing the flux outside the target as
traditional
collimator would. As the
title suggests, we have chosen to refer to
the
device as a reflecting collimator.
I
t is instructive to develop
a base case from which other designs can be evaluated
,
in
order to evaluate the
characteristics
of the collimation that our design requires.
A few
basic cases are analyzed
in this
section of the
report
. First the
analytical
response
of an
ideal detector
in a point source, target volume,
detector arrangement (Figure 1) is
considered. Then, MCNP is utilized to calculate the Reflection Factor F
R
(defined in
Appendix A) for
other geometries
. These are compared to the point source case,
revealing that no ba
sic collimator shape enhances the flux sufficiently (has great enough
F
R
) to permit rapid in
terrogation of a remote target. A quick
overview of
the cross
sections of known materials is presented to demonstrate the
absence of an ideal high
energy neutron r
eflecting material. Some advanced reflector concepts are presented
together with the theory behind their development. Finally, a quick outline of coming
research is presented.
The Glowing Cloud Analogy
The
order of magnitude calculation
s
discussed above
, executed in the following section,
quickly reveal
the n
ecessity of
a unique, highly efficient reflecting
collimator
design
.
In
order to realize such a
unique
design, it is necessary to develop a conceptual
understanding of high energy neutron transport
through reflecting media. A simple
a
nalogy provides an instructive mental picture.
Neutron transport in high Z materials
is
analogous to visible light transport in a diffuse scattering medium. In both cases
scattering is nearly isotropic in t
he laborato
ry
coordinate system and
the fraction of scattering
interactions which lead to absorption can be quite small
so
that
the number of scatters is quite large. Hence,
“
creating a collimator to reflect fast neutrons into a
beam is like making a flashlight refl
ector with a cloud.
”
From this analogy one can readily identify the nature of
the problem.
There
are no “surface effects” and
t
here is
no specular reflection for high energy neutrons.
The
next step is to develop a feel for the order of magnitude of th
e high energy neutron
scattering effects. This is the intent behind the baseline cases developed in the following
sections.
Baseline Cases
Point Source Calculation
The first baseline case is that of a point source and detector located 10m from the target.
The fraction of source particles that lea
d to a photon passing through
the detector
is
calcul
ated
.
The equations below describe the flux at distance,
r
, from
a
point source with
strength,
S
p
, incident on a target of volume, V, which presents interaction
coefficient µ
a
and yields
X(E)
photons per neutron interaction
. The Detector Reaction rate, R
d
,(photons per second)
encountered by an ideal photon detector presenting a cross
sectional area, A, to the target volume
is given as the final answer.
From this simple analysis we see that the collimator efficiency will be vital to the success
of a remote detection system
even with a source strength of 10
10
n/s.
A d
etection rate of
44 photons per second would require excessively lon
g detection times, failing an
important Q
FD requirement of fast response, indeed failing entirely to deliver a useful
signal, since the background photons from the interactions within the collimator would
likely hide this small target signal.
Planar Reflec
tor
The planar reflector is certainly not a concept that has be considered for implementation,
however, the reflection factor resulting from this arrangement helps to provide an
understanding for the affects of modifying other collimator geometries.
MCN
P was used to determine
the relationship between thickness of
the reflecting plane
and
the flux at the
target. From these values, the
reflection factor
F
R
was determined
(See Appendix A)
.
The relationship
between flux and distance
is illustrated
for seve
ral reflector thicnkesses
in
F
igure
6
below
.
Divergence of Neutron Field with
Graphite Planar Reflector
10 cm
y = 0.1161x
1.9839
20 cm
y = 0.1347x
1.9841
40cm
y = 0.1567x
1.9917
30 cm
y = 0.1481x
1.988
5 cm
y = 0.078x
1.9477
1 cm
y = 0.0797x
1.9864
0 cm
y = 0.0700x
1.9827
0.E+00
5.E08
1.E07
2.E07
2.E07
3.E07
3.E07
4.E07
4.E07
400
500
600
700
800
900
1000
1100
Source Target Distance (cm)
Flux (n/cm2*source particle)
A
trend line
has been fit to the data points for each curve. As expected, the field
diverges
near
ly
as
1/r
2
.
The ratio of Flux for the 40cm reflector to that with 0cm reflector is F
R
=2.23. The
most usefu
l information gathered from this simulation is the thickness of
reflector material that approximates an infinite medium. The mean free path (mfp) for 14
MeV neutrons in Graphite is about 6.4cm.
It seems that the flux at 6mfp
approaches
the
limit for an i
nfinite medium.
From this analysis, it is clear that other more efficient
reflector geometries must be developed.
Rectangular Collimator
Consider a basic collimator constructed form graphite, a strongly scattering but weakly
absorbing material used in nuc
lear reactor reflectors. As illustrated in Figure 4, the target
volume, a 50cm cube, is located 10m fr
om a point source
.
Collimator Diameter is 1m, with 50cm cubic rectangular cavity. The flux calculated
from MCNP simulations for varying Source Target
distances and reflector thicknesses
was plotted as in Figure XX for the Plane Reflector above. The result was quite similar
with the exception that reflection factor upper limit was 6.5 instead of less than 2.5.
However,
The reflection factor obtained f
rom this arrangement is significant, but far from
sufficient.
Other geometries or much larger collimator size will be required.
Search for Reflector Materials
The about 6.4cm mfp for neutrons in graphite necessitates the use of a very large
collimator.
Materials with 1/10 this mfp might enable the constr
uction of a portable
collimator. It is prudent to search for such a material.
The ideal reflector material would have the following characteristics.
1.
High scatter cross section at 14 MeV (
σ
S
>50 barns , actual 4)
2.
High Elastic Scatter to Other Scatter ratio
(
R
= σ
Elastic
/
σ
S
>100
, actual 3.5)
3.
High Atomic Number
(to limit thermalization, actual realizable)
4.
Inelastic Cutoff Above 6.5 MeV (doesn’t exist)
5.
Inexpensive
(realizable)
A quick look
at neutron cross sections for some common materials
reveals
that an ideal
material
does not exist.
A neutron
interaction cross section
from JENDL

3.3 (JAERI
2002)
presented in Figure 4 illustrates the
shortcomings of Iron as a typical material
.
Even materials like Cadmium, effectively “black” to thermal neutrons,
have a small
14MeV cross section. Lead and Tungsten are similar (see Appendix B).
14 MeV Cross
Section 3 barns
Resonance
Peaks 20 keV
Inelastic Cutoff
1 MeV
Capture Cross
Section
Advanced Collimator Geometries
Theory Guiding the Concepts
During the course of co
ncept generation, a few guiding questions were posed to direct the
efforts of the group towards developing a conceptual understanding of reflector design.
These include:
1.
Is it possible to create an ideal reflector
–
redirecting 100% of source particles t
o
target?
2.
If and ideal reflector exists, does it exist as a result of the dimensions being
stretched to infinity, or is there a finite version that reasonable approximates the
ideal case?
3.
Is there a way to determine whether a small change to a
given
reflec
tor shape will
enhance or decrease efficiency?
4.
Can the above method of determining
)
(
geom
f
F
R
be proceduralized and
converted to yield a useful computer program that can serve as a design aid?
5.
Is it possible to create a “white box
with a hole?
” Can a geometry serve to
contain neutrons, constantly randomizing their directions until these directions
align with the target, in which case escape probability becomes high and neutrons
leave toward the target?
The ultimate goal
is of course
to develo
p
the best
reflector geometry.
If an ideal
geometry exists, then its basic concept should be applied to known concepts to maximize
the efficiency of our design.
In such a geometry more than 50% of emitted neutrons are directed through the target. In
ord
er to achieve this there must be no leakage
in
2 dimensions in
the limit as dimensions
become infinite.
The guiding idea
that emerged was that an ideal collimating reflector would redirect
neutron current by minimizing
the
macroscopic cross section in
the direction of the
target, while maximizing cross section in other
direction.
This is concept seeks to answer the
white
box question
(#5). The answer
,
conceptually
,
is
that neutrons with directions other than toward
target will redirect, while those wi
th toward target
will not. It sounds elegantly simple.
The first geometry
contrived to meet
this criterion
was
a
two

dimensional stack of
scattering slabs
(Fig 7). The slabs
would alternate between
materials of
high and low cross section.
High Cross Section
Low Cross Section
Now, it is
a matter of proceduralizing the white box concept, so that this procedure can be
used to “prove” or disprove the geometry of our concepts.
Proceduralization
of the White Box Concept
The white box concept is currently being developed to serve as a theoreti
cal basis for
further reflector development. A brief outline of the theory is presented below.
A neutron’s
direction of travel can be described by 2 variables
,
with range
2
..
0
.
Say we chose
to select all neutro
ns that have direction
variables
in
a given range
(these
neutrons are selected to be sent towards the target)
.
We can then easily calculate
fraction
of scattering interactions which lead to a direction in the desired range
as
the fractional
solid angle of
the selected directions
4
S
F
,
assuming isotropic scattering (a good
behavioral first approximation). Given this fraction, we can determine the
spatial
distribution that neutrons should acquire in the scattering process
.
Basically, the
neutrons
will spread out in space more with increasing number of scatters
, flattening their spatial
distribution
. This implies that decreasing
F
will increase the size of the reflector in a
predictable manner.
It is believed that simple Diffusion Theory
can lend useful insight
into the nature of this spatial spreading out. At present, this concept has not been fully
developed; however, it is the current plan to fully develop a theoretical basis for
collimator design similar to what is described here.
S
hielding Considerations
The cross sections for Deuterium and Hydrogen are shown below. Deuterium has a
constant 3 barn cross section and is very conservative for thermal neutrons
The shielding structure has not been examined i
n detail to at this time. However, the
basic strategy used will probably be the classic approach. We will use water, heavy
water, or some wax
–
which ever material is found to thermalize over the shortest
distance
–
to slow neutrons into the thermal rang
e. Here, as many as possible will be
directed to the target, those that escape will be absorbed by Cadmium. Resulting gammas
will be attenuated by lead to limit operator dose and noise in nearby detector.
The Next Steps in Collimating Reflector Research
The direction of future research may take two very distinct
paths…
Thermalize and Collimate
–
Concentrate on Capture not Inelastic
This is the path that seems to lead to the best overall system, although sufficient analysis
has not be performed at this t
ime to say conclusively
Utilize a Huge Source/Shield Well
–
Enhance reflection as Possible
It may be possible to utilize a very high strength neutron source. This would make
efficiency of collimator less of a priority. However, shielding would have to b
e bulked
up to protect operators and bystanders…
Appendix A
Definitions
Reflection Factor
–
the definition of the reflection factor is quite simply
This definition permits quick comparison of a new geometry to already existing
geometries.
Ideal
Ref
lector
–
no leakage in any direction but the target direction. According the
reflecting light with cloud analogy, we would not see the collimator, unless we looked
down its axis
Appendix B
Cross Sections
In this appendix the cross sections for a few o
f the most promising reflecting materials
are presented. These materials are selected because of their relatively large 14MeV
elastic cross section. Unfortunately, Pb

208 is one of 4 stable isotopes. The others have
much higher capture cross sections an
d the inelastic cutoff is at lower energy. Otherwise,
lead, with its high density and reasonable cross section, might be used to construct a
reasonably small collimator.
14 MeV Mean Free Path (
MFP
)
Calculation
for Iron
cm
mfp
cm
mol
g
mol
atom
cm
g
atom
cm
M
N
s
A
s
s
89
.
5
1
169
.
0
/
8
.
55
/
10
022
.
6
)
/
87
.
7
(
/
10
2
1
23
3
2
24
Σχόλια 0
Συνδεθείτε για να κοινοποιήσετε σχόλιο