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