Photon Factory Activity Report 2005 #23 Part B (2006)
Instrumentation and Technique

4B2/2005G018

Correction of counting loss for X-ray detection system
Takashi IDA*, Yoshihiro IWATA, Hisashi HIBINO
CRL, Nagoya Institute of Technology, Asahigaoka 10-6-29, Tajimi, Gifu 507-0071, Japan
Introduction
The counting method is widely adopted in evaluation ofX-ray beam intensities. However, the effect of counting
losses due to the dead-time of the detector or the finite
response time of the detection circuits may cause serious
systematic errors in the evaluated intensities, especially athigh count rates.
Counting losses are usually modeled by a non-extended
or an extended dead-time model. The present study isintended to establish an improved method to determine
the dependence of the observed count rate on the true
count rate from the data measured by Chipman’s foil
method, and to correct the counting losses in the observed
beam intensity data, allowing deviation from the nonextended or extended dead-time models.
Models for counting losses
Intermediate model
An intermediate model between the non-extended andextended dead-time models can be constructed bysynthesizing the throughput functions for the two models
as follows [1]:n = f ex (f non− ex (r; τ 1 ); τ 2 ),where n is the observed count rate, r the true count rate,
f non− ex (r; τ 1 ) ≡ r (1 + rτ 1 )is the throughput function for the non-extended modelwith dead-time τ1, andf ex (r; τ 2 ) ≡ r exp(−rτ 2 )is the throughput function for the extended model withdead-time τ2.
It will be convenient to substitute the total dead-timeparameter τ and the degree of extension ρ for theparameters τ1 and τ2, via the equations:τ ≡ τ 1 + τ 2 , ρ ≡ τ 22 τ 2 ,n = f approx (r; τ , ρ) =exp(−r' t 2 )− exp(−2r' t 2 )t2r' = r (1 + rt1 ), t1 = τ − 3t 2 2 , t 2 = 6 ρ 13τ .
The maximum deviation of fapprox(r; τ, 1) from fex(r; τ) inthe range 0 ≤ r ≤ 1 τ is 0.0003 relative to the valuefex(r; τ) at r = 1 τ , which is smaller than the statistical
7errors predicted for data counts up to 10 . The inverse
function of the approximate function is given by theequations−1r = f approx(n; τ , ρ) = r' (1− r' t1 ),r' = −1 1 + 1− 4nt 2ln.
t22

Experimental validation
Chipman’s foil method is applied to the powder
diffraction intensity data of mica 003 reflection peak,measured with a high-resolution diffractometer located on
BL4B2. The observed intensities are well fitted byapplying the approximate intermediate model (Fig. 1),
while significant deviations have been found in fittingwith non-extended and extended models.

which gives the formula for the intermediate model:ªºρ rτ«».
f inter (r; τ , ρ ) =exp −«»1 + 1 − ρ rτ
¬ 1 + 1 − ρ rτ ¼(r)()

The formula exactly gives the non-extended and extended
dead-time dependence for ρ = 0 and ρ = 1, respectively.

Approximation for intermediate model
An approximate formula for the intermediate model [1],which has a simple expression of its inverse function, isgiven by

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Users' Report

Fig. 1. Results of fitting with the approximate
intermediate model, where y1 and y2 are attenuatedand unattenuated intensities. The experimental data
are shown as crosses and the optimized curve isshown as a solid line. The difference is shown as athick line and the estimated errors (±σ) are shown as
thin lines in the upper part.
Reference[1] T. Ida et al., J. Appl. Cryst. 84, 426 (2005).
* ida.takashi@nitech.ac.jp