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Journal ofApplied
Crystallography
ISSN 0021-8898

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Correction for counting losses in X-ray diffractometry
T. Ida and Y. Iwata

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J. Appl. Cryst. (2005). 38, 426–432

T. Ida and Y. Iwata ¯ Correction for counting losses

research papers
Journal ofApplied
Crystallography

Correction for counting losses in X-ray
diffractometry

ISSN 0021-8898

T. Ida* and Y. Iwata
Received 10 September 2004
Accepted 21 February 2005

# 2005 International Union of Crystallography
Printed in Great Britain – all rights reserved

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

Non-extended and extended dead-time models, a pulse-height analyser (PHA)
windowing model, and a model intermediate between the non-extended andextended models for losses in counting methods are compared. The validities ofthe methods are examined by application to the analysis of powder diffraction
peak intensity profiles measured by a foil method. The intermediate model
including parameters for the dead-time and degree of extension can reproduce
both the non-extended and the extended models and also intermediate
dependence between the two models. A convenient approximate formula for
the intermediate model, the maximum relative deviation of which is 0.0003, isalso proposed. The determination of the parameters and a correction for themeasured intensities can easily be achieved by applying the approximate model,
because it provides simple formulae for the correction function expressed as a
combination of elementary functions. Experimental and analytical methods for
precise evaluation of the parameters to specify the counting losses are alsopresented. Systematic deviations of the observed dependence from the nonextended and extended dead-time models have been detected by the precise
analyses of experimental data, while the PHA windowing model, intermediate
model and its approximation have reproduced the observed dependence within
the experimental errors.

1. Introduction
The counting method for evaluating diffracted beam intensity
is widely adopted in X-ray diffraction measurements. Detection methods with counting devices are often more favourable
for precise structure analyses because of the intrinsically linear
sensitivity and well defined nature of the statistical errors, incontrast to methods with time-integrating detectors, such as an
imaging plate (IP). However, the effect of counting losses dueto the dead-time of the detector or the finite response time ofthe detection circuits may cause serious systematic errors in
the intensities observed by the counting method, especially athigh count rates.
Counting losses are usually modelled by a non-extended
dead-time model or an extended dead-time model (Quintana,1991). In the non-extended dead-time model, it is assumed
that an event causes dead-time , but the following events
during the time  do not extend the dead-time. In the extended
dead-time model, the same event would extend the length ofthe dead-time by  starting at the arrival time of the uncounted
event.
The throughput function of the non-extended dead-timemodel (Müller, 1973) is given byn ¼ fnonex ðr; Þ ¼ r=ð1 þ rÞ;426

doi:10.1107/S0021889805005637

ð1Þ

where n is the observed count rate, r is the true count rate and is the dead-time. The solution of the above equation is givenby1
r ¼ fnonex
ðn; Þ ¼ n=ð1  nÞ;

ð2Þwhich can immediately be applied to correct the counting
losses.
The counting losses in a realistic detection system are more
likely to be modelled by the extended dead-time model(Omote, 1990; Reed, 1972), given byn ¼ fex ðr; Þ ¼ r expðrÞ:ð3Þ

Cousins (1994) has proposed the following formula forsynchrotron X-ray sources,
n ¼ Tb1 expðmrTb Þ½1  expðrTb Þ;ð4Þ

where Tb is the bunch interval and m is the largest integer not
greater than =Tb . For the usual case of   Tb, equation (4)can simply be reduced to the extended dead-time model.
The actual losses in the counting method are affected by thetime structure of the X-ray source and the operating characteristics of the detector/amplifier system. When a pulseheight analyser (PHA) or a single-channel analyser (SCA) is
used to reduce the effect of electrical noise, stray light or
higher-order harmonics of the monochromated incident beam,electronic reprint

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the settings for the PHA windowing may also affect the loss ofcounts (Cousins, 1994). Therefore, experimental evaluation ofcounting losses seems more practical than fully theoreticalprediction requiring many parameters to be determined.
Quintana (1991) has proposed a practical analytical method
to evaluate the dead-time parameter in the extended model,
from the experimental data measured by the foil method of
Chipman (1969), which can easily be performed with most
X-ray diffraction setups. However, the accuracy of thecorrection will be restricted if the actual counting losses
deviate from the exact extended model.
The present study is intended to establish an improved
method to determine the dependence of the observed countrate on the true count rate from the data measured by Chipman’s foil method, and to correct the counting losses in theobserved diffraction intensity data, allowing deviation fromthe non-extended or extended dead-time models. A convenient formula for correction, which can be used for intermediate dependence between non-extended and extended
dead-time models, is proposed.

2. Models for counting losses
2.1. PHA windowing model

When a pulse-height analyser is used to count the pulses,
the output count rate will be affected by the allowed pile-up
level L (Cousins, 1994). If all the input pulses are assumed tohave the same rectangular shape, the output count rate
through the PHA will be given byn ¼ r expð2rw ÞL1Pðrw Þ =l!;l

ð5Þwhich is identical to the extended dead-time model for ¼ 2w . In the L ! 1 limit, the detected count rate is
identical to the extended model for  ¼ w given byn ¼ r expðrw Þ:

The derivatives of the function at the origin (r = 0) show
singularities for some values of L as follows,
ðdn=drÞ0 ¼ 1;

n ¼ r expðrw Þ QðL; rw Þ;

ð6Þ

where the function Qð; zÞ is the Legendre incomplete gammafunction of the second kind, defined by11
R1 1R 1Qð; zÞ  t expðtÞ dt
t expðtÞ dt :
z

ð10Þ

8
< 1
ðd2 n=dr2 Þ0 ¼ 4w:
2w

ðL < 1ÞðL ¼ 1Þ ;ðL > 1Þ

ð11Þ

8
þ1
>
>
>
>
< 12w2
ðd3 n=dr3 Þ0 ¼ 1
>
>
0
>
>
: 2
3w

ðL < 1ÞðL ¼ 1Þð1 < L < 2Þ ;ðL ¼ 2ÞðL > 2Þ

ð12Þ

while the non-extended and extended dead-time models haveno singularities at the origin. The first, second and third
derivatives of the non-extended and extended dead-timemodels are given by0
fnonexð0; Þ ¼ 1;

ð13Þ

00
fnonexð0; Þ ¼ 2;

ð14Þ

000
ð0; Þ ¼ 6 2 ;fnonexð15Þ

fex0 ð0; Þ ¼ 1;

ð16Þ

fex00 ð0; Þ ¼ 2;ð17Þ

fex000 ð0; Þ ¼ 3 2 :

ð18Þ

and

l¼0

where w is the width of a pulse and L is the allowed pile-up
level normalized to the single-pulse-height level. The derivation of the formula is given in Appendix A. The above formula
can readily be extended for non-integer values of L by theequation

ð9Þ

The throughput of the PHA looks very much like theextended dead-time model, as can be seen in Fig. 1, eventhough there exist slight deviations for L 6¼ 1. Comparison of
the first few derivatives at the origin suggests that the deviation is in the direction of the non-extended model for
0 < L < 1 and in the opposite direction for 1 < L.

ð7Þ

0

Since the throughput function is specified by two continuous
parameters, w and L in equation (6), a usual nonlinear curve
fitting method can be applied to determine those parametersfrom the experimental data. The profiles of the throughput
function on variation of the allowed pile-up level L are shownin Fig. 1.
When the allowed pile-up level is equal to the single-pulseheight level (L = 1), the detected count rate is given by
Figure 1

n ¼ r expð2rw Þ;
J. Appl. Cryst. (2005). 38, 426–432

ð8Þ

Throughput predicted by a simple model for the effect of PHA
windowing.

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T. Ida and Y. Iwata  Correction for counting losses

427research papers
2.2. Intermediate model between non-extended andextended models

Table 1
Values of the first four derivatives of the counting loss model functions at
the origin.

An intermediate model between the non-extended andextended dead-time models can be constructed by synthesizing the two functions as follows:n ¼ fex ½fnonex ðr; 1 Þ; 2 ;

ð19Þ

where 1 and 2 are the dead-time parameters for the nonextended and extended components of the function, respectively. The first, second and third derivatives of the above
function at the origin are given by
ðdn=drÞ0 ¼ 1;

Orderf(x; , 0) = fnonex (x; )f(x; , 1)

fex(x; )

1
2
3
4

1
2
6 2
24 31
2
3 2
3.973 3

1
2
3 2
4 3

constructing the approximate formula is described inAppendix B.
The approximate model is given by the equationsn ¼ f ðr; ; Þ(t1 ½expðr0 t2 Þ  expð2r0 t2 Þ¼ 20rðd n=dr Þ0 ¼ 2ð1 þ 2 Þ;22

ðd n=dr Þ0 ¼ 612 þ 121 2 þ 322 ;3

3

while the derivatives of the component functions are given byequations (13)–(18). It will be convenient to substitute thetotal dead-time parameter  and the degree of extension  forthe parameters 1 and 2 , via the equations  1 þ 2 ;ð20Þ

  22 = 2 ;ð21Þ

which gives the formula
fintermediate ðr; ; Þ ¼ r½1 þ ð1  1=2 Þr1
 expf1=2 r½1 þ ð1  1=2 Þr1 g:

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T. Ida and Y. Iwata  Correction for counting losses

;

ð23Þ

r0 ¼ r=ð1 þ rt1 Þ;

ð24Þ

t1 ¼   3t2 =2;

ð25Þ

t2 ¼ ð6=13Þ1=2 ;ð26Þf ðr; ; 0Þ  fnonex ðr; Þ;ð27Þ

and also approximates the extended model for  = 1 reasonably well,f ðr; ; 1Þ ’ fex ðr; Þ:ð28Þ

The maximum deviation of f ðr; ; 1Þ from fex ðr; Þ in the range
0  r   1 is 0.0003 relative to the value of fex ðr; Þ at
r ¼  1 , which is smaller than the statistical errors predicted
for data counts up to 107.
The values of the derivatives of the functions f ðr; ; Þ,fnonex ðr; Þ and fex ðr; Þ at the origin are listed in Table 1.
The inverse function of the model function f ðx; ; Þ is givenby the equations
r ¼ f 1 ðn; ; Þ ¼ r0 =ð1  r0 t1 Þ;

2.3. Approximation for intermediate model

In order to evaluate the true count rate r from the observedcount rate n, the inverse function of the throughput function isneeded, while both the PHA windowing and the intermediate
models for counting losses described in xx2.1 and 2.2 lack
simple formulae for the inverse functions. Although thesolutions can be evaluated numerically by iterative calculation
based on Newton’s method (Cousins, 1994), the codings for
Newton’s method are often annoying, because the accuracy,
stability and efficiency of the iterative calculation strongly
depend on the initial guess of the solution, maximum iteration
times and criteria for convergence.
In this section, an approximate formula, which has a simple
expression of its inverse function, is proposed. The scheme for½t2 ¼ 0

where  is a parameter corresponding to the degree of deadtime extension, similarly to the exact intermediate model inthe preceding section.
This model is identical to the non-extended model for  = 0,
that is,ð22Þ

The formula exactly gives the non-extended and extended
dead-time dependences for  = 0 and  = 1, respectively. The
value of the third derivative of the throughput function at theorigin is linearly varied by changing the parameter  from 0
to 1.
Even if a sufficiently high count rate is not experimentally
available, or the statistical errors of the experimental data are
not small enough to estimate the precise value for the degreeof extension , the total dead-time parameter  can be estimated more safely, because it is determined by the initial
curvature of the throughput. The parameter  can easily betreated as a fixed parameter, if necessary.

½t2 6¼ 0

(
0

r ¼

t21 lnf½1 þ ð1  4nt2 Þ1=2 =2gn

½t2 6¼ 0
;½t2 ¼ 0

ð29Þ
ð30Þ

where t1 and t2 are related to  and  by equations (25) and(26).

3. Estimation of parameters3.1. Experimental

The experimental procedures referred to as Chipman’s(1969) foil method were applied to estimate the parameters in
the counting loss models. The 2 scan intensity profiles for the
003 reflection of mica powder (NIST, SRM675) were

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measured with a high-resolution powder diffractometer(Toraya et al., 1996) on beamline BL4B2 at the Photon Factory(PF) in Tsukuba. The 2 angle was measured with an encoder(Heidenhain RON-806) at each measurement step. The
specimen was rotated about its surface normal during themeasurements. One measurement was made with an aluminium foil absorber inserted in the beam path, and the second
measurement was made without the foil. A high-voltage/PHA
unit (Rigaku 5320 C1) was used to count the signal pulses. The
baseline and width of the window of the PHA were set to 0.5
and 2, respectively, relative to the first peak position in the
PHA differential curve. Changes in intensity caused by the
decay of the ring current were calibrated by monitoring theintensity of the incident beam.
3.2. Analytical method
3.2.1. Preliminary analysis. Since the 2 angles of the two

measurements did not strictly coincide, intensities expected
for the common 2 angles with equal intervals were calculated
by sampling intensity values from the curves drawn by a cubic
spline interpolation between the observed data points. The
first intensity data, fy1j g, were calculated from the data f21j g
and fY1j g measured with the foil absorber, and the second setof data, fy2j g, were calculated from the data f22j g and fY2j gmeasured without the foil.
Fig. 2 shows the observed intensity profiles for the first(attenuated) and second (unattenuated) measurements. Bothdata were measured at 0.001 steps in 2. The reduction in theobserved intensities near the peak top in the second
measurement clearly shows that it is heavily affected by losses
in the counting method.
The interpolated second set of intensity data fy2j g areplotted versus the interpolated first intensity data fy1j g in
Fig. 3. Firstly, the dependence was modelled by the extended
dead-time model, applying the fitting function
y01j ¼ fex ½a1 fex1 ðy1j ; Þ; ;

ð31Þ

where a is the transmittance of the foil.
The transmittance a and dead-time parameter  were
determined by applying a nonlinear least-squares optimization
to the observed dependence of fy2j g on fy1j g, to minimize thequantityPð32Þ
2 ¼ ðy2j  y01j Þ2 =j2 ;deviation showing a hysteresis-like behaviour is found in the
difference plot, which suggests a slight constant shift of theabscissa between the two measurements.
3.2.2. Correction of shift. In order to evaluate the shift of
the abscissa precisely, the correlation between the preliminarily modified values for the first data y01j , which were calculatedwith the optimized values of a and , and the second intensity
data set y2j was examined. The correlation was calculated by
ck ¼

nk1
P

y01j y2;jþk ;

ð34Þ

j¼0

where n is the total number of data points. A periodicity ofckþn ¼ ck

ð35Þ

is assumed.
Fig. 4 shows the correlation curve and results of fitting witha Lorentzian function:
fLor ð2Þ ¼ b þ ðS=wÞ½1 þ ð2  20 Þ2 =w2 1 ;ð36Þ

Figure 2
The peak profiles of the mica 003 reflection measured at the Photon
Factory. The profile measured with a foil absorber is shown as a thin line,
and that measured without the foil is shown as a thick line.

jwhere j is the estimated error for each data point. The errors
were estimated by assuming counting statistics for the second
intensity data set fy2j g, given byj2 ¼ ðy2j =T2j Þ1=2 ;ð33Þwhere T2j is the measurement time for the second data set at
each step, for the preliminary evaluation of the parameters.
The inverse function fex1 ðx; Þ was calculated by a Newton
method, with the initial guess of the solution as x=ð1  xÞ,
ten-times iteration and no criteria for convergence.
The result of the preliminary fitting to the data with theextended dead-time model is also shown in Fig. 3. Systematic
J. Appl. Cryst. (2005). 38, 426–432

Figure 3
The result of a preliminary fitting with the extended dead-time model tothe data shown in Fig. 2. The experimental data are shown as crosses andthe optimized curve is shown as a solid line in the lower part of the figure.
The difference is shown as a thick line and the estimated errors ( ) are
shown as thin lines in the upper part.

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Table 2
Results of optimization for different counting loss models.
a is the transmittance of the foil,  is the dead-time, L is the allowed pile-up
level and  is the degree of dead-time extension.

Figure 4
Correlation curve to evaluate the slight shift. The experimental
correlations are shown as dots and the optimized curve is shown as athin solid line in the lower part. The upper part shows the difference plot.

Figure 5
The result of the final optimization with the extended dead-time modelusing the data shown in Fig. 2. Data are represented as described in thecaption of Fig. 3.

where b is the background, S is the intensity, 20 is the peakshift and w is the half width at half-maximum of the Lorentzian function.
The optimized value for the peak shift is 20 = 0.00025 ,which means that the second set of data are shifted by 0.00025
to the lower-angle side as compared with the first data set.
The shift-corrected intensity data for the second measurement, fy02j g, were then created by applying a cubic splineinterpolation to the data set shifted by 20, f22j þ 20 g and
fY2j g.
3.2.3. Final optimization. Final optimization was conductedto minimize the quantityP2 ¼ ðy02j  y01j Þ2 =ðj0 Þ2 ;ð37Þ
jwhere j0 is the estimated error for each data point. The
statistical errors in the first measurement are also taken into
account in the final optimization, applying the equation
j0 ¼ ½ y02j ðT2j1 þ T1j1 y01j =y1j Þ1=2 ;ð38Þwhere T1j and T2j are the measurement time at each step for
the first and second data set, respectively.

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T. Ida and Y. Iwata  Correction for counting losses

Model

a

 (s)

ExtendedNon-extended
PHA-windowing
Intermediate
Approximate

0.03807 (3)0.03639 (3)0.03768 (6)0.03784 (4)0.03784 (4)0.803 (1)1.196 (2)0.364 (4)0.842 (5)0.842 (5) or L2

L = 0.900 (11)
 = 0.872 (15)
 = 0.872 (15)260.83548.4
194.2
187.8
187.8

Fig. 5 shows the result of the final fitting analysis with theextended dead-time model, where the shift correction of0.00025 is applied to the data shown in Fig. 2. The statisticalerrors corresponding to 1 standard deviation are shown as
thin lines in the upper part of Fig. 5. The difference plot,shown as thick lines in the upper part of Fig. 5, suggests slightsystematic deviation of the observed dependence from thatpredicted by the extended dead-time model. It should be
noted that such slight systematic deviation could not be
detected without the shift correction of 0.00025 , which ismuch smaller than the measurement step of 0.001 .
3.2.4. Comparison of models for counting losses. The nonextended dead-time model, the PHA-windowing model in
x2.1, the intermediate model in x2.2 and the approximate
intermediate model in x2.3 were also applied to the data
analysed in the preceding section. The results for the fourmodels are shown in Figs. 6–9 and are listed in Table 2.
The non-extended dead-time model poorly fits the observed
dependence (Fig. 6). However, the tendency of the systematicdeviation is found to be in the opposite direction to that of theextended dead-time model shown in Fig. 5. This result suggeststhat the observed dependence has a character intermediate
between the non-extended and extended dead-time models.
When the allowed pile-up level L is treated as an adjustableparameter, the PHA-windowing model can fit the observed
dependence reasonably well (Fig. 7). There remain no significant systematic deviations in the difference plot of Fig. 7.
However, the optimized value L = 0.900 (11) is inconsistent
with the PHA settings applied for the measurement, where thebaseline and width of the window are set to 0.5 and 2 relative
to the first peak position in the PHA differential curve. Thereseems no theoretical evidence to suggest any advantage of thesimple PHA-windowing model for counting losses. The optimized value of L less than 1 means that the deviation from theextended dead-time model (L = 1) is in the direction of the
non-extended model, as discussed in x2.1.
The intermediate model gives slightly better fits (smaller 2 )to the observed dependence, as shown in Fig. 8. The optimized
value  = 0.872 (15) shows that the experimental dependence
is mainly reproduced by the extended dead-time model( = 1), but the contribution of the non-extended dead-timemodel ( = 0) is not negligible.
Application of the approximate intermediate model,
described in x2.3, gives no difference from the exact intermediate model within the experimental errors (Fig. 9), exceptthat the codings are easier and the computation time needed iselectronic reprint

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Figure 6

Figure 8

The result of the final optimization with the non-extended dead-timemodel using the data shown in Fig. 2. Data are represented as described in
the caption of Fig. 3.

The result of the final optimization with the intermediate model using thedata shown in Fig. 2. Data are represented as described in the caption ofFig. 3.

Figure 7

Figure 9

The result of the final optimization with the PHA-windowing model using
the data shown in Fig. 2. Data are represented as described in the captionof Fig. 3.

The result of the final fitting with the approximation for the intermediate
model using the data shown in Fig. 2. Data are represented as described in
the caption of Fig. 3.

shorter in the approximate model, where no iterative calculation is needed for evaluation of the inverse function. Itshould be noted that the efficiency in the evaluation of theinverse function will become more significant when the model
is applied to correct the counting losses in experimental
powder X-ray diffraction intensity data with many data points.

APPENDIX A
Derivation of a counting loss model for PHA
windowing
It is assumed that the input pulses have a common rectangularshape with width w . The Poisson distribution for the numberof pulses N in a fixed time T is given byfP ðN; TÞ ¼ ðrTÞN expðrTÞ=N!;

4. Conclusion
Non-extended and extended dead-time models, a PHAwindowing model, an intermediate model, and an approximate intermediate model for counting losses have been
applied to high-precision analyses of experimental diffraction
peak intensity profiles measured by a foil method. The
deviations of the experimental dependence from the convenient approximate model are within the experimental errors.
Both the optimization of the parameters and corrections forcounting losses can easily be achieved by applying theapproximate model, because the exact solution of its inverse
function is expressed by a combination of elementary functions.
J. Appl. Cryst. (2005). 38, 426–432

ð39Þ

where r is the average count rate.
When the PHA is set to allow only single-height
pulses (L = 1) and to reject any pile-up peaks, an input
pulse will be detected only when the interval from the last
pulse and the interval to the next pulse are both longer
than the pulse width w . The probability that each pulse isdetected is given by expð2rw Þ, because the probability thatno pulse occurs within the period w is given byfP ð0; w Þ ¼ expðrw Þ. The output count rate is then expected
to be r expð2rw Þ.
For the case of L = 2, the following logic is applied. (i) A
pulse is temporarily detected when the interval from the last
pulse is longer than w . (ii) The temporarily detected pulse iscancelled when two or more pulses occur within the following

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T. Ida and Y. Iwata  Correction for counting losses

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period w , or fixed otherwise. Then the probability that eachpulse is detected is given byfP ð0; w Þ½fP ð0; w Þ þ fP ð1; w Þ ¼ ð1 þ rw Þ expð2rw Þ; ð40Þ
which predicts the output count rate to be
n ¼ rð1 þ rw Þ expð2rw Þ:
Similarly, the probability that each pulse is detected for thegeneral case of the allowed pile-up levels L is given byfP ð0; w Þ

L1
P

fP ðl; w Þ ¼ expð2rw Þ

l¼0

L1
P

ðrw Þl =l!;

ð41Þ

l¼0

It is suggested that the formula is close to the extended deadtime model but slightly deviated, to have a weak non-extended
character, because the value 28 2 =9 is close to but slightly
greater than the third derivative of the extended model3 2 , while that of the non-extended dead-time model is givenby 6 2.
It is also suggested that the extended dead-time model canbe approximated by mixing the formulae of the non-extended
dead-time model, if a negative value of the dead-time parameter for the non-extended component is allowed. We assume
a mixed function asf ðr; t1 ; t2 Þ ¼ f0 ½fnonex ðr; t1 Þ; t2 ;ð52Þ

f0 ðr; t2 Þ ¼ t21 ½expðrt2 Þ  expð2rt2 Þ:ð53Þ

and the output count rate is given byn ¼ r expð2rw ÞL1Pðrw Þl =l!;

ð42Þ

l¼0

where the level L is normalized to the single-pulse-heightlevel.

The first, second and third derivatives of the mixed function at
the origin are given by

APPENDIX B
Scheme for constructing an approximate formula
Let us examine the formulan ¼ f0 ðr; tÞ ¼ t1 ½expðrtÞ  expð2rtÞ;
which is equivalent
X ¼ expðrtÞ,

to

the

quadratic

equationX 2  X þ nt ¼ 0:

f 0 ð0; t1 ; t2 Þ ¼ 1;

ð54Þ

f 00 ð0; t1 ; t2 Þ ¼ 2t1  3t2 ;ð55Þ

f 000 ð0; t1 ; t2 Þ ¼ 7t22 þ 18t1 t2 þ 6t12 :

ð56Þ

ð43Þ

Substituting the solution of f 00 ð0; t1 ; t2 Þ ¼ 2 for t1 in equation (56) gives

forf 000 ð0; t1 ; t2 Þ ¼ 6 2  13t22 =2:

ð44Þ

The solution is immediately given by
X ¼ expðrtÞ ¼ ½1 þ ð1  4ntÞ1=2 =2;

ð45Þr ¼ t1 lnf½1 þ ð1  4ntÞ1=2 =2g:

ð46Þ

Since the first, second and third derivatives of the formula at
the origin are given by
ðdn=drÞ0 ¼ 1;

ð47Þ

ðd2 n=dr2 Þ0 ¼ 3t;

ð48Þ

ð57Þ

The first, second and third derivatives of the mixed function at
the origin then become identical to those of the extended
dead-time model for t2 ¼ ð6=13Þ1=2 . When we introduce
another parameter, , to satisfy the relation
t2 ¼ ð6=13Þ1=2 ;
the third derivative of the mixed function becomes linearly
dependent on .
The present study was financially supported by the Nippon
Sheet Glass Foundation for Materials Science and Engineering. The authors are indebted to the staff of the Photon
Factory for their aid and for making the facilities available.

Referencesðd n=dr Þ0 ¼ 7t ;
3

3

ð49Þ

2

the formula becomes comparable to the non-extended andextended dead-time models for t ¼ 2=3, which gives432

ðd2 n=dr2 Þ0 ¼ 2;

ð50Þ

ðd3 n=dr3 Þ0 ¼ 28 2 =9:

ð51Þ

T. Ida and Y. Iwata  Correction for counting losses

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