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