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Journal ofApplied
Crystallography
ISSN 0021-8898

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Monte Carlo simulation of the effect of counting losses on measured
X-ray intensities
T. IdaCopyright © International Union of Crystallography
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J. Appl. Cryst. (2007). 40, 964–965

T. Ida ¯ Monte Carlo simulation

short communications
Monte Carlo simulation of the effect of counting
losses on measured X-ray intensities

Journal ofApplied
Crystallography
ISSN 0021-8898

T. IdaReceived 14 June 2007
Accepted 6 August 2007

Ceramics Research Laboratory, Nagoya Institute of Technology, Asahigaoka, Tajimi 507-0071, Japan.
Correspondence e-mail: ida.takashi@nitech.ac.jp# 2007 International Union of Crystallography
Printed in Singapore – all rights reserved

The statistical properties of intensities affected by counting loss based onconventional non-extended and extended dead-time models are examined by aMonte Carlo method. It has been conﬁrmed that the variance of the counted
pulses for the non-extended dead-time model with the rate of generated pulses
2¼ non =ð1 þ rÞ2, while that for ther and the dead-time  is given by non2
¼ ext ½1  2r expðrÞ, as proposed
extended dead-time model is given by ext
by Laundy & Collins [(2003). J. Synchrotron Rad. 10, 214–218], for the meanvalues of counted pulses non and ext , respectively. Practical formulae toestimate the statistical errors of the corrected intensities are also presented.

1. Introduction
Counting methods are widely used to measure the intensity of X-rays.
Independently generated signal pulses are expected to obey thePoisson distribution, which predicts that the possibility for thenumber of pulses n during the measurement period T is given by
PPoisson ðnÞ ¼ ðn!Þ1 ðrTÞn expðrTÞ;
where r is the average rate of generated pulses. The mean andvariance of the Poisson distribution are simply given by2Poisson ¼ Poisson¼ rT:

Therefore, the statistical errors of the intensity measured by acounting method can naturally be modelled by the square root of themeasured count, if a Poisson distribution of the pulses is assumed.
However, it is not expected that the intensity measured with arealistic counting system should strictly obey the Poisson distribution,
because it is always affected by the counting loss caused by the ﬁnite
response time of a detector and/or the electronic circuits in thedetection system.
The probability for the number of counted pulses m based on thenon-extended dead-time model can be expressed asPnon ðmÞ ¼ ðm!Þ1 ð1 þ rÞ½rðT  mÞm exp½rðT  mÞ;

ð1Þ

where  is the dead-time. Details about the derivation and validationof the above formula will be discussed elsewhere. The mean andvariance of the above distribution are approximately given bynon ¼ rT=ð1 þ rÞ;

ð2Þ2
¼ non =ð1 þ rÞ2 :
non

ð3Þ

Recently, Laundy & Collins (2003) have reported analytical formulae
for the statistical properties of a pulse-overlap model for the counting
loss, which includes the conventional extended dead-time model as aspecial case. According to their results, the mean and variance for theextended dead-time model are given by964

ext ¼ rT expðrÞ;ð4Þ2¼ ext ½1  2r expðrÞ:
ext

ð5Þ

It should be noted that the variance of the number of counted pulses
is expected to be smaller than the mean value in both the nonextended and the extended dead-time models, while it should coincide with the mean in the case of the Poisson distribution.
Since all the formulae given in equations (2)–(5) are derived as thesolution for the limit T  , it will be worth examining the validityof the application to the case of a ﬁnite ratio of T to .
In this communication, the statistics of the non-extended andextended dead-time models are examined by applying a Monte Carlo
method with a simple algorithm, and compared with the theoreticalvalues calculated from equations (2)–(5). The statistical errors in thecorrected intensity estimated by the conventional models of counting
loss are also discussed.

2. Simulation
A simulation of conventional models for counting loss applied topulses that obey the Poisson distribution is easily implemented asfollows: (i) a pulse is generated at the time t ¼ t0 þ t after a random
interval t from the time t0 when the last pulse is generated, wheret is calculated by t ¼ ðln xÞ=r from a random number x evenly
distributed between 0 and 1; (ii) the pulse is counted when t  t0 > in the case of the extended dead-time, and t  tv > , where tv is thearrival time of the last counted pulse, in the case of the non-extended
dead-time model; (iii) t0 is replaced by t, and tv is also replaced by twhen the pulse is counted. Processes (i)–(iii) are repeated while t < T.
Fig. 1 plots the mean and variance of 1000 trials of the nonextended dead-time simulations for various rates of pulse r. The
measurement period and the dead-time are ﬁxed at T ¼ 1 and ¼ 0:001. The results of the simulation are well modelled by thecurves given by equations (2) and (3) up to r ’  1.
The corrected intensity of each non-extended dead-time simulation is calculated byci ¼ mi =ð1  mi =TÞ;ð6Þwhere mi is the number of counted pulses. The variance of thecorrected intensity fci g and the corresponding variance curve derived
from equation (2), given bydoi:10.1107/S002188980703854X

electronic reprint

J. Appl. Cryst. (2007). 40, 964–965

short communications

Figure 1

Figure 2

Mean and variance of the non-extended dead-time simulation and the theoreticaldependence on variation of the average rate of pulses obeying Poisson statistics.

Mean and variance of the extended dead-time simulation and the theoreticaldependence on variation of the average rate of pulses obeying Poisson statistics.

 2 ¼ rTð1 þ rÞ;

ð7Þ

are also plotted in Fig. 1. Even though the simulated statistical errorsof the corrected intensity rapidly increase when the rate of pulse r
approaches  1, the errors are reasonably modelled by applying
equation (7).
Fig. 2 plots the results for the extended dead-time simulations andcorresponding theoretical curves. The parameters used in theextended dead-time simulation are the same as those for the nonextended dead-time simulation. The results of simulation are wellmodelled by the theoretical curves calculated by equations (4) and(5), again up to r ’  1.
The corrected intensity data fci g for the extended dead-time
simulation are calculated by applying the following approximation(Ida & Iwata, 2005):

t2 ¼ ð6=13Þ1=2 :
The variance of the corrected intensity from the extended dead-time
simulation is well modelled by the variance curve given by (Laundy &Collins, 2003)
 2 ¼ rTð1  rÞ2 ½expðrÞ  2r;ð8Þ

as shown in Fig. 2.
In conclusion, the statistical errors of the corrected intensity, whichshould properly be taken into account in least-squares or maximumlikelihood analysis (Antoniadis & Berruyer, 1990), can be estimatedby applying equation (7) or (8).

ci ¼ r00 T=ð1  r00 t1 Þ;
r00 ¼ t21 lnf½1 þ ð1  4mi t2 =TÞ1=2 =2g;
t1 ¼ ½1  3ð6=13Þ =2;
1=2J. Appl. Cryst. (2007). 40, 964–965

References
Antoniadis, A., Berruyer, J. & Filhol, A. (1990). Acta Cryst. A46, 692–711.
Ida, T. & Iwata, Y. (2005). J. Appl. Cryst. 38, 426–432.
Laundy, D. & Collins, S. (2003). J. Synchrotron Rad. 10, 214–218.

T. Ida  Monte Carlo simulationelectronic reprint965