Post on 05-Jan-2016
description
Average path length for neutrons from point source into cylindrical detector
Jenny Williams, UoC
The problem
• 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)
50cm
4.5cmNeutronpointsource
60.5cm
Path length through cylinder
• Need to find average length of all possible paths through cylinder - average of dotted linesaverage 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.
r
{dr
For each annulus, calculate:number paths X length of pathThis is equivalent to:area of annulus X length of pathNormalise by dividing by the total number of paths – i.e. divide by the area of the endface of the detector.
Detector front endface
Calculation of average path length
2
02)(
R
rdrrfR
r is radial distance from cylinder centre where neutron enters cylinderR is radius of cylinderf(r) is path length through cylinder for a neutron entering cylinder at radial distance r from centreline.
Path length at r times area of annulus at r
Total area of cylinder endface
Non-uniform flux at cylinder endfaceFlux 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 cmReasonable approximation to flux non-uniformity
Average path distance formula becomes:
55.5 cm
From Steffen’s BRM talk – agenda of 27 June
R
R
rdr
rdrrf
rh
rh
0
0
2
2)(
)(
)(
Where h(r) is a Gaussian fitted to the55cm beam profile to represent the non-uniform flux of neutrons entering the detector
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 cm27.936 cm