Average path length for neutrons from point source into cylindrical
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Transcript Average path length for neutrons from point source into cylindrical
Average path length for neutrons
from point source into
cylindrical detector
Jenny Williams, UoC
The problem
60.5cm
Neutron
point
source
4.5cm
50cm
• Neutrons from approx point source into cylindrical
detector
• Want to know average path length of a neutron through
the detector
• As a first example, we’ll assume a uniform flux of
neutrons from the point source (flux-included calculation
follows)
Path length through cylinder
• Need to find average length of all possible paths
through cylinder - average of dotted lines
• Slightly different formula for neutrons exiting
back endface (easy to deal with)
Total number of neutron paths
through cylindrical detector
• Number of paths proportional to area of front endface the
neutrons go through
• Path length, f(r), is a function of radial distance r from
centreline that neutrons enter the detector
– 50cm at r =0, and 0cm at r =4.5cm
• At radial distance r, number of paths is proportional to
area contained in annulus of radius r, and thickness dr.
For each annulus, calculate:
number paths X length of path
This is equivalent to:
area of annulus X length of path
Normalise by dividing by the total
number of paths
– i.e. divide by the area of the
endface of the detector.
r
dr
Detector
front
endface
Calculation of average path length
R
0
f (r )2 rdr
R
2
Path length at r times area of
annulus at r
Total area of cylinder endface
r is radial distance from cylinder centre where neutron enters cylinder
R is radius of cylinder
f(r) is path length through cylinder for a neutron entering cylinder at
radial distance r from centreline.
Non-uniform flux at cylinder endface
Flux profile takes account of source
not being entirely point-like
Have flux profile at 55cm
– small attenuation with r
Beam enters cylinder at 60.5 cm
Reasonable approximation to flux
non-uniformity
55.5 cm
Average path distance formula
becomes:
R
f (r ) h(r )2 rdr
0
R
0
h(r )2 rdr
Where h(r) is a Gaussian fitted to the
55cm beam profile to represent the
non-uniform flux of neutrons entering
the detector
From
Steffen’s
BRM talk –
agenda of
27 June
Average path length calculations
• Can calculate uniform flux path length
analytically
• Including non-uniform flux distribution
need numerical integral
– Easy to do with ROOT
• Comparing analytic and numerical results
for uniform case provides a useful cross
check.
Average path length – the NUMBER
• Average path traversed by neutrons from
a point source passing through cylindrical
detector:
• For detector of radius 4.5cm, length 50cm,
at distance 60.5cm from neutron source:
27.936 cm