research papers
Statistical properties of measured X-ray intensities
affected by counting loss

Journal ofApplied
Crystallography
ISSN 0021-8898

T. IdaReceived 15 July 2008
Accepted 24 September 2008

# 2008 International Union of Crystallography
Printed in Singapore – all rights reserved

Ceramics Research Laboratory, Nagoya Institute of Technology, Asahigaoka, Tajimi, 507-0071,
Japan. Correspondence e-mail: ida.takashi@nitech.ac.jp

The statistical properties of X-ray intensities measured with counting systems
have been experimentally investigated. A formula of statistical variance for theintermediately extended dead-time model is proposed and compared with theexperimentally evaluated variance obtained from repeated measurements basedon Chipman’s foil method applied to X-ray detection systems of laboratory andsynchrotron powder diffractometers. It has been found that the variance of theobserved intensities is smaller than the average of count, as has been suggested
by conventional theoretical models for counting loss. It is shown that thestatistical errors can be predicted by applying an intermediately extended deadtime model including dead-time  and degree of extension  as fixed parameters.

1. Introduction
The weighted least-squares method is widely applied toanalysis of diffraction intensity data. The sum of the squared
deviation of each observation from the calculated value can betreated as a maximum-likelihood estimator, where the deviation is weighted by the reciprocal of the known statistical error
in the measurement.
It is usually assumed that the statistical error of the X-ray
intensity measured by a counting method is identical to thesquare root of the measured count, which can be justified if thestatistical distribution of the counted pulses obeys Poisson
statistics. Independently generated signal pulses are certainly
expected to obey the Poisson distribution, which predicts thatthe probability of measuring a number of pulses n during themeasurement period T is given by
PPoisson ðnÞ ¼ ðn!Þ1 ðrTÞn expðrTÞ;

ð1Þ

where r is the average rate of generated pulses. The mean
2of the Poisson distribution are
Poisson and variance Poisson
2¼ rT.
simply given by Poisson ¼ Poisson
However, it is not expected that the intensities measured
with a realistic counting system should strictly obey the
Poisson distribution, because they are always affected bycounting loss caused by the finite response time of a detector
and/or the electronic circuits in the detection system (Omote,1990).
The mean and variance of the counted pulses based on the
non-extended dead-time model with a dead-time  areapproximately given by (Müler, 1974)
non ¼ rT=ð1 þ rÞ;

ð2Þ2
¼ non =ð1 þ rÞ2 ;
non

ð3Þ

J. Appl. Cryst. (2008). 41, 1019–1023

while those predicted by the extended dead-time model are
given by
ext ¼ rT expðrÞ;ð4Þ2¼ ext ½1  2r expðrÞ:
ext

ð5Þ

The validity of the above approximate formulae has recently
been confirmed by a systematic investigation based on Monte
Carlo simulations (Ida, 2007). This study suggests that thestatistical errors of the observed count and the statisticalerrors to be attached to the corrected intensity data can beevaluated if a detection system is modelled by either the nonextended or the extended dead-time model.
However, no reasonable formulae to predict statisticalvariance for an intermediately extended dead-time model (Ida
& Iwata, 2005) have been reported, despite the fact that it canmodel experimental behaviours measured with the detailed
Chipman (1969) foil method more flexibly than the conventional dead-time models.
In the present study, a hypothetical formula of statisticalvariance for the intermediately extended dead-time model is
proposed and compared with the experimental statisticalproperties obtained from repeated measurements based on
the Chipman foil method applied to X-ray detection systemsof laboratory X-ray and synchrotron powder diffractometers.

2. Intermediately extended dead-time model
The intermediately extended dead-time model was originallydeveloped to model the intermediate character of the countloss behaviour observed for a real X-ray detection system (Ida
& Iwata, 2005). The average count int is given byint ¼ fext ½ fnon ðr; 1 Þ; 2 T;

ð6Þ

where
doi:10.1107/S0021889808030732

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fnon ðr; Þ ¼ r=ð1 þ rÞð7Þ

fext ðr; Þ ¼ r expðrÞ

ð8Þ

and

are throughput functions for the non-extended and extended
dead-time models, respectively.
It is convenient to substitute the total dead-time parameter and the degree of extension  for the parameters 1 and 2 ,via the equations  1 þ 2 ;

ð9Þ  22 = 2 ;ð10Þ

so that the formula exactly gives the non-extended andextended dead-time dependences for  ¼ 0 and  ¼ 1,
respectively. The throughput function of the model is thenexpressed by
fint ðr; ; Þ ¼ fext ½ fnon ðr; 1 Þ; 2 ;

ð11Þ

1 ¼   2 ;
2 ¼ 1=2 :It is also known that the formula given by Ida & Iwata(2005),
 1
t ½expðr1 t2 Þ  expð2r1 t2 Þ ½t2 6¼ 0
fint0 ðr; ; Þ ¼ 2
r1½t2 ¼ 0;ð12Þ
r1 ¼ r=ð1 þ rt1 Þ;
t1 ¼   3t2 =2;
t2 ¼ ð6=13Þ1=2 ;is a good approximation for fint ðr; ; Þ, and an explicit formulafor the inverse function of fint0 ðr; ; Þ is available as1fint
0 ðm; ; Þ ¼ r1 =ð1  r1 t1 Þ;


r1 ¼



t21 ln ½1 þ ð1  4mt2 Þ1=2 =2m½t2 6¼ 0
½t2 ¼ 0;

ð13Þ
ð14Þ

where m is the observed count rate.
It should be noted that the formula for fint ðr; ; Þ given inequation (11) is not simply the model for a series of two
counting devices with non-extended and extended dead-timecharacters, because the output pulses affected by the nonextended dead-time counting loss cannot be modelled by
Poisson statistics, while the formula of equation (8) implicitly
assumes the Poisson statistics of the input pulses for theextended dead-time device.
Changing the viewpoint, the formula given by equation (11)would be justified for a hypothetical series of two devices,where the throughput of the first device is given by equation(7) but the output pulses from the first device are still
expressed by Poisson statistics, and the second device seriallyconnected to the first device loses counts following the1020

T. Ida  Statistical properties of measured X-ray intensities

extended dead-time scheme given by equation (8). Then thestatistical variance of the intensity measured by the hypothetical series is logically given by2
int¼ r2 T expðr2 2 Þ ½1  2r2 2 expðr2 2 Þ;

ð15Þ

r2 ¼ r=ð1 þ r1 Þ:

ð16Þ

Even though the above formula of variance for the intermediately extended dead-time model may appear too artificial, it will still be worth trying to compare the values
predicted by the model with experimental observations.

3. Experimental
3.1. Laboratory X-ray diffractometer

The counting-loss characteristics of the X-ray detection
system of a laboratory powder X-ray diffractometer (Rigaku
RAD-2C) were investigated. A Cu K tube operated at 40 kV
and 30 mA was used as the X-ray source. Several aluminium
foils were inserted into the beam path to adjust the maximum
intensity, which was measured with a scintillation counter. The
allowed level of the signal pulses was restricted with a pulse
height analyser (PHA) unit to within 26 and 166% relative tothe most frequently detected level of pulses.
A home-made mechanism to insert and remove an additional attenuator of nine aluminium foils with 20 mm thickness(180 mm thickness in total) was attached to the diffractometer;
this attenuator was remotely operated by the controller for thediffractometer.
A flat single-crystalline plate of -quartz (001) was placed at
the specimen position, and the X-ray intensity was varied in 48
steps by finely scanning the 2/ axis around the 003 reflection
peak located at 2 ’ 26:7 .
At each intensity step, two sets of measurements of X-ray
intensity were obtained, each over a 0.5 s period, repeated 500
times by inserting and 500 times by removing the additional
attenuator. This means that the number of unattenuated andattenuated intensity data at each intensity step is N ¼ 1000 foreach condition. The total measurement time needed tocomplete the data collection was about 14 h. The timedependence of the measured count observed at a certain
intensity data point is shown in Fig. 1.
3.2. Synchrotron powder diffractometer

The counting-loss characteristics of an X-ray detection
system of a synchrotron powder diffractometer with multiple
detection systems on beamline BL-4B2 at the Photon Factoryin Tsukuba (Toraya et al., 1996) were also investigated.
A synchrotron beam monochromated at 0.12 nm with adouble Si(111) crystal monochromator and collimated with acylindrical mirror was used as the X-ray source. Since theelectron storage ring was operated in multi-bunch mode
during the measurement, the time structure of the synchrotron
X-rays is not expected to be significant for the detection
system, which has a response time of about 1 ms. The direct
beam attenuated with a molybdenum foil with 10 mm thickness
J. Appl. Cryst. (2008). 41, 1019–1023

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was introduced onto a scintillation counter. A mechanism toinsert and remove an additional attenuator of 16 Al foils with
12 mm thickness (192 mm thickness in total) was attached on
the upstream side on the entrance slit (2.5 mm width, 0.5 mm
height) of the diffractometer. The intensity of the X-rays at thedetector was varied in 41 steps by rotating stepwise a centre
slit of 0.5 mm in height attached to the  axis. The allowed
level of the signal pulses was restricted with a PHA (Rigaku
5320 C1) to within 50 and 150% relative to the mostfrequently detected level of pulses.
At each intensity step, five sets of measurements of X-ray
intensity were obtained, each over a 0.5 s period, repeated 100
times by inserting and 100 times by removing the additional

attenuator. The total measurement time needed to complete
the data collection was about 12 h. The time dependence ofthe measured count observed at a certain intensity data pointis shown in Fig. 2.

4. Results and discussions
4.1. Analysis of data collected with laboratory diffractometer

Since the number of counts observed for the laboratory
diffractometer has shown slight time dependence, as can beseen in Fig. 1, a segmented linear dependence has been fitted
to the observed intensities, and the variance of residuals is
treated as the modified statistical variance of the measured
intensity. No significant time dependence has been observed in
the values of residuals.
The mean, variance of raw data and variance of residuals
for the intensities measured on removal of the additional
attenuator (unattenuated intensities) are plotted versus theaverage count measured on insertion of the additional
attenuator (attenuated intensities) in Fig. 3.
The values of mean m, variance of raw data v and varianceof residuals s for the intensity data sets fyi g ði ¼ 1; . . . ; NÞ are,respectively, calculated as
m ¼ N 1

N
P

yi ;

ð17Þ

i¼1

v¼

N
P

ðyi  mÞ2 =ðN  1Þ;

ð18Þ

i¼1

Figure 1
Typical time dependence of intensities measured by the repeated
Chipman method for the laboratory powder X-ray diffractometer. (a)Attenuated intensities (grey dots) and fitted segmented linear dependence (solid lines), (b) residuals of the fit shown in (a), (c) unattenuatedintensities (grey dots) and fitted segmented linear dependence (solidlines), and (d ) residuals of the fit shown in (c).

Figure 3

Figure 2
Typical time dependence of intensities measured by Chipman measurements for the synchrotron powder diffractometer. See Fig. 1 fordefinitions.
J. Appl. Cryst. (2008). 41, 1019–1023

Results of repeated Chipman measurements for the laboratory powder
X-ray diffractometer. Lower panel: average unattenuated intensities
mhigh (crosses), variance of raw data vhigh (open circles), variance ofresiduals shigh (solid circles), fitting curve for mhigh based on anintermediately extended dead-time model (solid line), and variancecalculated with the optimized parameters (broken line) plotted versus
average attenuated intensities mlow. Upper panel: residuals of fitting(crosses) and statistical errors about the average (solid lines, see text),
and the errors estimated for a single Chipman measurement (brokenlines).
T. Ida  Statistical properties of measured X-ray intensities1021

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s¼

NP½yi  gðti ; a1 ; . . . ; aP Þ2 =ðN  PÞ;

ð19Þ

i¼1

where gðt; a1 ; . . . ; aP Þ is a function with P parameters to
optimize the segmented linear dependence on the time t.
The errors for the above expectation values are, respectively, estimated by
X
1=2
N
ðyi  mÞ2
;
ð20Þ
m ¼
NðN  1Þ
i¼1
v ¼

X
N
ðyi  mÞ4
i¼1

s ¼


X
N
i¼1

N3 2
v
NðN  1Þ

1=2
;

ð21Þ

4
1=2
yi  gðti ; a1 ; . . . aP Þ
s2

:
N2
N

ð22Þ

N2



The dependence of the mean unattenuated intensities mhighupon the attenuated intensities mlow is fitted by using anapproximate formula for the intermediately extended deadtime model (Ida & Iwata, 2005), given by
1mhigh ¼ fint0 ½ fint
0 ðmlow =T; ; Þ=a; ; T;

ð23Þ

where a is the transmittance of the additional attenuator.
The observed dependence of mhigh versus mlow is fairly well
reproduced by the model with the optimized values of transmittance a ¼ 0:11806 ð12Þ, dead-time  ¼ 1:706 ð2Þ ms anddegree of extension  ¼ 0:6450 ð16Þ.
It is also concluded that the dependence is not completely
modelled by equation (23), as the residuals plotted as crosses
in the upper panel of Fig. 3 show a systematic deviation larger
than the estimated statistical errors of the average count,drawn as solid lines (which are almost coincident with the zeroline owing to their small values). The errors of intensities
estimated for a single Chipman measurement, simply calculated by multiplication of the errors for the average byðN  1Þ1=2 ¼ 9991=2 , are shown as broken lines in the upper
panel of Fig. 3. The figure demonstrates that the incomplete-Figure 4
Results of repeated Chipman measurements for the synchrotron powder
diffractometer. See Fig. 3 for definitions.

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T. Ida  Statistical properties of measured X-ray intensities

ness of the model would not appear significant if datacollected by a single Chipman measurement were used toevaluate the parameters a,  and  in equation (23).
The plot of the observed variance, vhigh and shigh , in Fig. 3
clearly indicates that the variance evaluated by the repeatedmeasurements is systematically smaller than the average mhigh,as has been predicted by the conventional theoretical models
for counting loss of detection systems (Müler, 1974).
The curve calculated by equation (15) with the values of a, and  optimized to fit the mean values is drawn as a broken
line in Fig. 3. The observed values of variance are slightly
larger than the values calculated by the intermediately
extended dead-time model, which appears more significant in
the higher count rate regions. The slight difference still seems
reasonable, because the model does not include any otherorigins than the segmented linear time dependence for thevariation of the source X-ray intensities.
4.2. Analysis of data collected with synchrotron diffractometer

The number of counts observed for the synchrotron
diffractometer has shown a more significant time dependence
than that for the laboratory diffractometer, as can be seen in
Fig. 2. The data shown in Fig. 2 were measured at the highest
intensity point in the series of measurements, and the amplitude of the periodic time dependence was about 0.05% relative to the average value. The gradual decrease in intensity
may be attributed to the decay of the synchrotron beam
supplied from the storage ring, while the origin of the periodic
behaviour is not clear. A segmented linear dependence has
been fitted to the observed intensities, and the variance ofresiduals is treated as the modified statistical variance ofmeasured intensity, similarly to the analysis of the datacollected with the laboratory diffractometer. No significant
time dependence could be found in the values of residuals.
The mean mhigh , variance of raw data vhigh and variance ofresiduals shigh of the intensities measured on removal of theadditional attenuator (unattenuated intensities) are plottedversus the average count mlow measured on insertion of theadditional attenuator (attenuated intensities) in Fig. 4.
The dependence of the mean unattenuated intensities mhighon the attenuated intensities mlow is well fitted by equation(23), with the optimized parameters of transmittancea ¼ 0:12276 ð3Þ, dead-time  ¼ 0:9470 ð4Þ ms and degree ofextension  ¼ 1:0521 ð9Þ. The observed dependence is notcompletely modelled by equation (23), as the residuals plottedin the upper panel of Fig. 4 show systematic deviation larger
than the statistical errors, similarly to the results for thelaboratory diffractometer.
The dependence of raw variance vhigh on mlow shows a large
dispersion, as can be seen at the highest intensity point in
Fig. 4, while the dependence of shigh is found to be almost
smooth within the estimated errors shigh. This figure suggeststhat most of the periodic time dependence shown in Fig. 2 hasbeen removed by subtracting the segmented linear dependence.
J. Appl. Cryst. (2008). 41, 1019–1023

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Table 1

Table 2

True count rate r, observed count rate robs (both in counts per second),
variance of observed count ðrobs TÞ2 and variance of corrected countðrc TÞ2 , predicted for the laboratory diffractometer with a dead-time
 ¼ 1:706 ms and degree of extension  ¼ 0:6450.

Count rate (counts per second) and variance predicted for thesynchrotron diffractometer with a dead-time  ¼ 0:947 ms and degree ofextension  ¼ 1:052.

r

robs

ðrobs TÞ2ðrc TÞ2

300
1000
3000
1  104
3  1041  105
3  105

300
998
2985
0.983  104
2.852  104
0.847  104
1.876  105

300
996
2960
0.957  104
2.629  104
0.651  105
0.912  105

300
1002
3022
1.024  104
3.226  104
1.286  105
7.194  105

The curve calculated by equation (15) with the values of a, and  optimized to fit the mean values is drawn as a broken
line in Fig. 4. Even though the experimental variances havebeen estimated at larger values than the calculated curve, thevariance of residuals shigh approaches closer to the calculated
curve than the raw variance vhigh. Since the time-dependent
variation of the source X-ray intensities is not likely to be fullymodelled by the segmented linear dependence, it is suggested
that the statistical variance calculated by equation (15) still
gives an appropriate description of the statistical variance ofthe measured count.
4.3. Correction of counting loss

When the dead-time  and the degree of extension  are
known, the corrected count rate rc can be calculated from theobserved count rate robs ¼ m=T by applying the following
equation:
1
rc ¼ fint
0 ðrobs ; ; Þ:

ð24Þ

Statistical errors for the corrected count rate, rc , can becalculated byrc ¼ robs ðdrc =drobs Þ ¼ robs ðdrobs =drc Þ1 ;ð25Þ

ðrobs Þ2 ¼ r2 T 1 expðr2 2 Þ ½1  2r2 2 expðr2 2 Þ;

ð26Þ

r

robs

ðrobs TÞ2ðrc TÞ2

300
1000
3000
1  104
3  1041  105
3  105

300
999
2991
0.991  104
2.916  104
0.909  104
2.253  105

300
997
2974
0.972  104
2.751  104
0.749  105
1.267  105

300
1001
3008
1.009  104
3.085  104
1.106  105
4.434  105

calculated using the parameters estimated in xx4.1 and 4.2 forthe laboratory and synchrotron diffractometers.
The statistical variance of the corrected intensity ðrc TÞ2 isclose to the average count in the low count rate region,
whereas it shows a rapid increase when the true count rate r
approaches the reciprocal of the dead-time  of the detection
system. This result indicates that the correction for statisticalerrors is necessary on application of the weighted leastsquares method as a maximum-likelihood estimation, especially in the high count rate region.

5. Conclusion
The statistical properties of X-ray intensities measured withlaboratory and synchrotron powder diffractometers have been
experimentally investigated. Repeated application of Chipman’s method has revealed that the statistical variance of theobserved intensity, affected by counting loss of the detection
system, is lower than the average, as has been suggested byconventional theoretical models for counting loss.
The statistical errors can be evaluated for any observed
intensity by applying the intermediately extended dead-timemodel including dead-time  and degree of extension  as
fixed parameters.

where

r2 ¼ rc =ð1 þ rc 1 Þ;ð27Þ

drobs =drc ¼ ðdr2 =drc Þ ð1  r2 2 Þ expðr2 2 Þ;

ð28Þ

dr2 =drc ¼ 1=ð1 þ rc 1 Þ2 :

ð29ÞIt should be noted that the error rc for the corrected countrate rc diverges to infinity when the count rate approaches theinverse of the dead time  1.
Tables 1 and 2 list the expected observed count rate, robs ,and the variances of the observed and corrected intensities,ðrobs TÞ2 and ðrc TÞ2 , for several values of true count rate r,

J. Appl. Cryst. (2008). 41, 1019–1023

Part of this work has been performed under the approval of
the Photon Factory Program Advisory Committee (proposal
No. 2007 G093).

References
Chipman, D. R. (1969). Acta Cryst. A25, 209–214.
Ida, T. (2007). J. Appl. Cryst. 40, 964–965.
Ida, T. & Iwata, Y. (2005). J. Appl. Cryst. 38, 426–432.
Müler, J. W. (1974). Nucl. Instrum. Methods, 117 401–404.
Omote, K. (1990). Nucl. Instrum. Methods Phys. Res. Sect. A, 293,
582–588.
Toraya, H., Hibino, H. & Ohsumi, K. (1996). J. Synchrotron Rad. 3,
75–83.

T. Ida  Statistical properties of measured X-ray intensities1023