５７Comparison of radon doublet alpha spectra
measured in water using chi-squareand maximum likelihood methods
Shunichi Hamanaka*, Kiyoshi Shizuma** and Wen Xiao-qiong***
Abstract
Radon doublet alpha spectra were measured in water using the chi-square and maximum
likelihood methods and the measurement results were compared. Radon measurements were madewith a high-resolution alpha liquid scintillation spectrometer ORDELA 8100AB. The chi-square
method introduced significant systematic deviations in cases of low-count statistics whereas themaximum likelihood method yielded good results in all cases. Thus, radon concentration in water canbe determined more accurately using the maximum likelihood method.
1. Introduction
Precise estimates of radon (２２２Rn) concentrations indrinking water and ground water have become
necessary in recent years. These estimates are basedon measurements of alpha and/or gamma rays ofradon and its daughter nuclides. We have recently
developed a alpha liquid scintillation spectrometricmethod using the ORDELA 8100AB spectrometer［１］. In
this method, alpha rays of radon are measured
directly, before radioactive equilibrium has beenreached between radon and its daughters. Since thealpha peaks of ２２２Rn and its daughter nuclide ２１８Po
overlap, the radon peak area is determined by peakcurve fitting, which has traditionally been done using
the chi-square method because of its simplicity.
Recently, the maximum likelihood method has become
popular for decay curve fitting［２］. Several reports havenoted that systematic deviations are introduced when
the chi-square method is applied to radioactive data,
which usually follow a Poisson distribution. This isespecially true with low-count statistics, whereas themaximum likelihood method gives good results［３−５］ overa wide range of radioactive counts.
In the present work, double Gaussian fitting witha linear background for the alpha doublet of ２２２Rn 5.49
MeV and ２１８Po 6.00 MeV was carried out to obtain theradon peak area by means of the chi-square method
and the maximum likelihood method. We investigatedmainly the estimation of parameter values (known as“point estimation” in statistics) and compared theresults to determine the radon concentration in water
accurately. Some examples of radon measurement inwater are given.
2. Experimental
Radon in 1 L of sample water was extracted into 3
mL of liquid scintillator consisting of 4g of 2,5diphenyloxazole, PPO, dissolved in 1 L of xylene, andthen 1 mL of the scintillator was transferred to a 1
mL glass counting vial. Alpha rays with an energy of
5.49 MeV emitted from the decay of radon in the vial

Fig. 1 Cross-section of the detector portion of the alphaliquid scintillation spectrometer ORDELA 8100
AB. One mL of liquid scintillator is put into a 1
mL glass counting vial.

＊General Education＊＊Applied Nuclear Physics, Faculty of Engineering, Hiroshima University＊＊＊Institute of Modern Physics, Chinese Academy of Sciences

平成１３年８月２４日受理５８弓削商船高等専門学校 紀要 第２４号（平成１４年）

pressed as follows : 
$ (1)The first and second terms express the ２２２Rn and
２１８
Po peaks. The third and fourth terms express thebackground part. The quantities  are the freeparameters that would be searched in the analysis.
The parameters ,  and ,  represent the peakheights and peak positions. The parameter  isconnected to the peak FWHM , which is a common
value for ２２２Rn and ２１８Po peaks, by the relation :Fig. 2 Typical alpha liquid scintillation spectrum of radonand its daughters for 3600 sec measuring time.
Radon concentration in water sampled at OnomichiTrackstation was 2311 BqL−１.

were directly measured with a high-resolution alphaliquid scintillation spectrometer ORDELA 8100AB,shown in Fig. 1［６］. This method was described indetail previously［１］. The typical energy spectrum ofradon and its daughters in ground water having ahigh radon concentration is shown in Fig. 2. The
alpha peaks of ２２２Rn 5.49 MeV and daughters ２１８Po 6.00
MeV and ２１４Po 7.69 MeV were superimposed on thebeta and gamma continuum of the daughters ２１４Pb and２１４Bi. In the present work, the ground water was
sampled at Kagamiyama (where the radon concentration is about 26 BqL−１）, near Hiroshima University.
Radon measurements were made for 60 minutes oncea day for 20 days to provide doublet alpha spectra for
analysis.

+ (   $3.1 Peak-shape function of alpha doublet
Since the peak of a single-energy alpha ray
obtained with the detector system in the present work
is essentially symmetrical and is described by aGaussian curve［６］, the doublet of ２２２Rn and ２１８Po fully
dissolved in the liquid scintillator is described by adouble Gaussian curve of the same peak width. Fig. 2shows how the ２１４Po 7.69 MeV peak spreads slightly tothe low-energy side due to a wall effect. That is
because some of the ２１４Pb and ２１４Bi has adhered to thewall of the vial before the ２１４Po emitted its alpha ray［７］.
Beta-gamma background is represented by a straightline in the region of the alpha peaks. Therefore, thepeak-shape function for this alpha doublet has theform of two Gaussian curves plus a linear function forthe channel number and can be ex- %$$$%

(2)



The parameters
andrepresent the intercept
and slope of the background part. Consequently, theradon peak area is given as+  !$(3)3.2 Maximum likelihood method
Since the number of counts in each channel
follows the Poisson distribution, the probability toobtain the number of counts in thechannel  is given by %$$$%

(4) %where  is the parent mean of . Then thelikelihood function
3. Analysis(is expressed as* 
(5)
 The best fitted could be obtained when thislikelihood function is maximized.
Usually, it isconvenient to use the natural logarithm of thelikelihood function expressed as follows : ) (6)The optimum values of the free parameters  canbe obtained by the following simultaneous equations,which are likelihood equations :' % %$$$%
'(7)

Since the log likelihood function
is non-linearwith respect to , an iterative fitting procedureComparison of radon doublet alpha spectra measured in water using chi-square and maximum likelihood methods（ 中・静間・温）５９

similar to ref. 3 for the linearization was carried out.
The initial values of the free parameters  wereobtained on MCA’s display to guarantee fast convergence and get reliable results. Parameter searchingfound astopped when the log likelihood function
maximum. The uncertainties of parameters wereapproximately derived from the error matrix［８］. Thismethod is denoted by MXL.

the spectrum data. The background was represented
by a straight line, which was determined by averaging
the low- and high-energy sides of the doublet. The
weight was calculated from the original data in thesame way as in LSQ1. This method is denoted by
LSQ2.

3.3 Two chi-square methods
The chi-square method is widely used to analyze
the nuclear radiation spectrum assuming a normal
distribution with the same value of the parent mean
and the parent variance according to eq. (4). The
quantity chi-square is defined byFig. 3 shows typical results for curve fitting. Whenthe statistics are good, as in Fig. 3(a), the threedifferent analytical methods agree well on which is thebest-fitted function, except the small background in
LSQ1 due to the pull-down effect. When the statistics
are poor as in Fig. 3(b), the pull-down effect makes theFWHM of the chi-square fit smaller than that of the
maximum-likelihood fit and sometimes reduced thepeak height. The radon peak area is therefore
underestimated in LSQ2, while LSQ1 may yield acloser estimate of the peak area because the pull-down
effect shifts the background downward.
It was confirmed that MXL preserves total counts
in the fitting region and the free parameters are(8)# ) %where  is the weight for the datum . The mostprobable values for free parameters are determined byminimizing the chi-square. Generally, the weight  isgiven by &", where " is the standard deviation forthe datum 
. Since the parent variance is not known,

the weight is deduced in various ways［３，９］. In thepresent work Neyman’s chi-square is adopted replacing  by
. When the value of  is zero,  istaken as unity so as to prevent the weight being
infinite.
The chi-square method suffers from potential
problems, especially when the data consists of lowcount statistics. One of these problems is due to thediscrepancy between the Poisson and normal distributions. It seems that agreement is good as long as thenumber of counts is above 5［１０］, but Y. Jading and K.
Riisager have shown that this is not true fordetermination of the parameter value［５］. Anotherproblem is due to the weighting used in the fitting.
For Neyman’s chi-square, since the weights are large
for datum points over the fitted curve and small fordatum points under the curve, the fitting function ispulled down and the fitted area is underestimated.
This is referred to as the “pull-down effect”, asdescribed below.
The chi-square function is non-linear with respectto , so a linearization method similar to MXL wascarried out. This method is denoted by LSQ1.
Moreover, the pull-down effect causes the simultaneous fitting of peak and background to give smaller
background values, so we subtracted the backgroundand performed double Gaussian chi-square fitting for &

4. Results and discussion

Fig. 3 Typical results for curve fitting for doublet alpha
spectra of radon. The solid lines represent thebest-fitted functions by MXL, the dashed lines
ones by LSQ1 and the dotted lines ones by LSQ2 :(a) third-day spectrum ; (b) eighteenth-day spectrum.

６０

弓削商船高等専門学校 紀要 第２４号（平成１４年）

Fig. 4 Differences in background obtained by threedifferent methods as a function of elapsed time.
Each point indicates the difference in backgroundcounts per channel between maximum likelihood
fit and two chi-square fits, calculated by averaging the background in the region of the radon
peak.

unbiased. The differences of background determined
by LSQ1, LSQ2, and MXL are shown in more detail inFig. 4 as a function of elapsed time corresponding tofrom 13.4 to 3.4 (counts/channel). It was found thatthe difference between LSQ1 and MXL is approximately unity, which is consistent with the result of ref.
5. It was also found that the results of LSQ2 and MXL
almost agree. These findings mean that the background term includes most of the bias in LSQ1 andthe discrepancy of the peak term becomes large inLSQ2.
Ratios of the radon peak area of two chi-square
fits to MXL are shown in Fig. 5 as a function ofelapsed time corresponding to from 2486 to 96 (counts)of radon peak area ( =102.5〜4.7) determined byMXL. It may be stated that :a) the result of LSQ2 is always underestimated dueto the pull-down effect,b) it deviated beyond the error limit after thetwelfth day with below 263 (counts) of radon peakarea (  10.9),c) the result of LSQ1 is close to that of MXL
containing the background reduction problem,d) the result of LSQ1 is sometimes underestimatedin poor statistics because of the strong pull-down
effect of the radon peak.



5. Application

Fig. 5 Ratios of radon peak area determined by threedifferent analysis methods as a function of elapsedtime. Each point indicates the ratio of radon peakarea of chi-square fit to MXL.

Radon concentration in water can be calculatedusing the effective efficiency which is determined
experimentally for radon in 1 mL of scintillator［１］.
Some examples of radon concentration measurement
are given in Table 1. Radon concentrations determined by three different methods almost agree in thecases of Nos. 1 and 2, but the deviations between them
are significant in the case of No. 4. A reliable
estimate of radon concentration in water can be madeby means of the maximum likelihood analysis of theTable 1 Radon concentration in ground water obtained by alpha−liquid scintillation spectrometry and analyzed bythree different analysis methods.
No.

PlaceRn peak count＊(sec)(counts)(counts)

MXL

LSQ1

LSQ2

Meas. time

２２２＊

Radon concentration (BqL−１）

1

Innoshima

3600

5998

232.2

75.3１±1.9１

75.6１±1.9１

75.2１±1.9１

2Kagamiyama

3600

2035

１88.8

26.3１±1.0１

26.6１±1.0１

26.0１±1.1１

3Yuge

3600

1393

１56.5

17.9１±0.8１

18.1１±0.8１

17.3１±0.8１

4Taguchi

3600

１162

１１6.1

１1.98±0.36１1.59±0.32

１1.01±0.35＊) 222)Rn peak count and peak height  are obtained by MXL.

Comparison of radon doublet alpha spectra measured in water using chi-square and maximum likelihood methods（ 中・静間・温）radon alpha doublet.
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