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Journal of Applied Mathematics and Physics, 2021, 9, 15221540
https://www.scirp.org/journal/jamp
ISSN Online: 23274379
ISSN Print: 23274352
DOI:
10.4236/jamp.2021.97104 Jul. 22, 2021 1522
Journal of Applied Mathematics and Physics
Monte Carlo Simulation Study of HotParticle
Detection in Voluminous Samples by Gamma
Spectrometry
Liang T. Chu1,2, Adam G. Burn1,2, Clayton J. Bradt1, Thomas M. Semkow1,2
1Wadsworth Center, New York State Department of Health, Albany, NY, USA
2Department of Environmental Health Sciences, School of Public Health, University at Albany, State University of New York,
Rensselaer, NY, USA
Abstract
In this work, we addressed the inhomogeneity problem in gamma spectro
metry caused by hot particles, which are dispersed into environment
from
large nuclear reactor accidents such as at Chernobyl and Fukushima. Using
Monte Carlo simulation, we have determined the response of a gamma spec
trometer to individual and grouped hot
particles randomly distributed in a
soil matrix of 1L and 0.6
L sample containers. By exploring the fact that the
peaktototal ratio of efficiencies in gamma spectrometry is an empirical pa
rameter, we derived and verified a powerlaw relationship betwee
n the peak
efficiency and peaktototal ratio. This enabled creation of a novel
calibration
model which was demonstrated to reduce the bias range and bias standard
deviation, caused by measuring hot particles, by several times,
as compared
with the homogeneous calibration. The new
model is independent of the
number, location,
and distribution of hot particles in the samples. In this
work, we demonstrated successful performance of the model for a single
peak
137Cs radionuclide. An extension to multipeak radionuclide was also derived.
Keywords
Chernobyl, Fukushima, Peak Efficiency, Total Efficiency,
Signal Detection
Theory
1. Introduction
In gamma spectrometry of environmental, food, and industrial matrices, volu
minous samples are usually analyzed in quantities ranging from a fraction of to
several L or kg. This is done in order to increase the sensitivity as well as to bet
How to cite this paper:
Chu, L.T.,
Burn,
A.G.,
Bradt, C.J. and Semkow, T.M. (2021
)
Monte Carlo Simulation Study of Hot

Particle
Detection in Vol
uminous Sam
ples by Gamma
Spectrometry
.
Journal of Applied Mathema
t
ics and Phy
sics
,
9
, 15221540.
https://doi.org/10.4236/jamp.2021.97104
Received:
June 9, 2021
Accepted:
July 19, 2021
Published:
July 22, 2021
Copyright © 20
21 by author(s) and
Scientific
Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution
International
License (CC BY
4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
L. T. Chu et al.
DOI:
10.4236/jamp.2021.97104 1523
Journal of Applied Mathematics and Physics
ter assess the extent of any contamination present. Germanium gamma detec
tor (Ge) calibration for voluminous samples is accomplished with either the
physical traceable standards or computational methods. Both types of calibra
tions assume homogeneous distribution of radioactivity in large samples, using
bulk peak efficiency
pb
of the Ge detector. In many types of samples, however,
distribution of radioactivity may be heterogeneous. This can lead to substantial
bias in activity determination.
One type of inhomogeneity may be referred to as geometrical, where different
sections of the sample may have varied radionuclide activities. This has been in
vestigated for spiked reference materials [1], where analysis of variance was used
to determine homogeneity. Assumptions about geometrical inhomogeneity were
studied in terms of cylinder and disc [2], fraction of volume not containing radi
ation [3], or two sections of Marinelli beaker [4] [5].
Another type of inhomogeneity can arise from sample granularity. It was
shown, using MonteCarlo (MC) simulation, that the Ge detector peak efficiency
dropped with increased granularity [6]. However, this effect has not been seen
experimentally for grains of soil or polystyrene much smaller than the container
size, in which spiking solution occupied an interstitial space [7].
Yet another important type of inhomogeneity of interest to this investigation
is due to suspension of “hot particles” in the sample matrix. The problem of hot
particles was originated from nuclear detonation fallout [8] and was extensively
observed and studied following the 1986 Chernobyl [9] [10] and 2011 Fukushi
ma nuclear accidents [11] [12]. Hot particles from nuclear accidents are the
results of several formation mechanisms: disintegration of nuclear fuel in the ex
plosion and fires; condensation of liquid droplets; and deposition of volatile fis
sion products such as Cs on fly ash and atmospheric aerosols [13]. The composi
tion of hot particles depends on the formation mechanism. The Chernobyl par
ticles can be monoelemental, bielemental, or fuel fragments [13], whereas Fu
kushima particles contain predominately 137Cs, referred to as Cs microparticles
[12]. Hot particles can have diameters from a fraction of a micrometer to over
100 μm [14] and their size distribution is often assumed as lognormal [15] or ar
bitrary [16].
Hot particles travel significant distances with the plume [17] [18]. Gasphase
and aerosol 131I from Fukushima were observed as far as New York State [19].
Hot particles deposit on the ground and cause contamination of soil, water,
crops, food, etc. Additional mechanisms for hot particle dispersion from nuclear
accidents involve weathering of radioactive lava (melted fuel elements) [20] and
sorption of ionic 137Cs on soil particles [21].
An entirely different source of hot particles, also referred to as discrete ra
dioactive particles, is corrosion of neutron irradiated steel in normal nuclear
reactor operation [22] as well as fuel reprocessing [23]. The presence of hot par
ticles in potential dirty bomb explosions has been described [24].
The presence of hot particles in voluminous samples creates unusual chal
L. T. Chu et al.
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Journal of Applied Mathematics and Physics
lenges in gamma spectrometry regardless of their origin. There are several ap
proaches to ameliorate these challenges, such as instrumental, radiochemical,
and modeling. On the instrumental side, one can perform digital radiography to
identify locations of hot particles [25]. Rotating waste drum scanning techniques
can locate and determine heterogeneous distribution of radioactivity, utilizing
emission/transmission measurements combined with modeling [26] [27] [28]
[29] [30].
Among radiochemical methods, mechanical mixing only repositions hot par
ticles in the sample without homogenization. Chemical homogenization of large
samples is difficult. In addition, the particles formed as fuel elements, by con
densation, or explosion are typically refractory and only hightemperature fu
sion [31] or HF digestion [32] can dissolve them. However, the ionic fraction,
such as containing aerosoldeposited Cs and I, can be homogenized for some
matrices as has been demonstrated for food using tetramethylammonium hy
droxide and enzymes [33] [34].
A voluminous sample can contain both uniformly distributed radioactivity as
well as inhomogeneous hot particles. It was estimated that at least 65% of total
activity in the 30km zone around Chernobyl was due to hot particles [14]. Sta
tistical modeling methods have been developed based on either splitting of a
large sample and measurement of several subsamples on a gamma spectrometer
[15] [35], or repetitive mixing and measuring of the same sample [36] [37] [38].
By analyzing the variance and modeling, some information about the fraction of
activity in hot particles, their number, and size distribution could be inferred.
It follows from this introduction that a single gamma spectrometric mea
surement of a voluminous sample containing hot particles can be biased,
i.e.
, the
measured activity in the sample can significantly differ from true activity. By
using techniques of sample splitting or mixing followed by repetitive measure
ments, that bias can be reduced, however, the resulting dispersion can be large.
In environmental health risk assessment, it is most important to obtain as ac
curate and precise determination of activity as possible. Therefore, the aim of
this investigation was to reduce bias and improve precision, when measuring
voluminous samples containing hot particles. We are seeking a novel gamma Ge
detector calibration model in a functional form of
( )
p fq=
, where detector
peak efficiency
p
is a function of empirical parameter
q
. In this model, we are
aiming at finding
p
which better represents sample inhomogeneity than the bulk
efficiency
pb
, thereby reducing the bias and dispersion of measurement.
The methodology in this paper is by MC simulation, following [39] [40], and
is described in detail in Section 2. This study focuses on a single radionuclide
137Cs, which is the most prominent gamma emitter remaining several years after
the nuclear accident [36]. We use both the gamma radiation detected as a peak
at 661.66 keV as well as Comptonscattered radiation. Scattered radiation has
been used extensively in measuring void fraction using gamma radiation [41]
[42] [43], medical image reconstruction [44], and geometrical inhomogeneity
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Journal of Applied Mathematics and Physics
[5].
In gamma spectrometry, the most significant measures of Ge detector perfor
mance are peak efficiency
p
as well as the total efficiency
t
, the latter being a sum of
peak and Compton radiations [45]. The ratio of peak efficiency to total efficiency
p
/
t
is larger for larger Ge crystals and for those with Comptonsuppression sys
tems. This ratio is used in analytical approaches to coincidencesumming correc
tions [46]. Some authors prefer using its inverse,
t
/
p
[47]. The
p
/
t
ratio is a di
rectly measurable quantity, at least for a single radionuclide source, and it is a
strong function of gamma energy. It has been demonstrated that
p
/
t
also de
pends on the radioactive source position with respect to the Ge detector [48], the
feature we explore in the present work.
We design two calibration models: a 1particle model and an
n
particle mod
el. For the 1particle model, the relation of
( )
p f pt=
is derived using gam
ma attenuation [49], and its dependence on calibration is described in Section 3.
The
n
particle model is described in Section 4. The effect of
p
vs.
p
/
t
is more
complicated for this model and has to be interpolated between those for single
particles and bulk sample efficiency. We describe the interpolation process us
ing Signal Detection Theory (SDT) [50] in Section 4. The performance of the
n
particle model, when activities of the particles are not equal, and the extension
of the model to multipeak radionuclide are described in Section 5, followed by
discussion in Section 6 and conclusions in Section 7. In this work, we are not
considering radiation counting statistics and focus exclusively on the dispersion
caused by inhomogeneity.
2. Monte Carlo Simulations
All calculations were performed for the 661.66keV gamma ray from 137Cs, using
sand as sample matrix with measured density of 1.55 g∙cm−3. Two counting geo
metries were considered as depicted in Figure 1: 0.6L and 1L cylindrical con
tainers. Two coaxial ptype Ge detectors were used, with efficiencies of 134% and
48% relative to a 7.6cm (3inch) by 7.6cm sodium iodide detector, which are also
depicted in Figure 1. The dimensions in Figure 1 are to scale, whereas the actual
values are given in Table 1. These configurations are existing in our laboratory.
They can be compared in terms of detectability, which is defined as a ratio of the
sample volume to the Ge crystal volume for the same activity in both samples. The
configuration 0.6 L/134% Ge has such detectability 4.8 times better than the 1
L/48% Ge configuration. However, when the samples are assumed as having the
same specific activity, the detectability is the product of sample volume and the
Ge crystal volume. Then, such detectability is respectively 1.7 times better.
The calculations were performed using the MC code Gespecor, version 4.2
[39]. This program is especially designed for calculations in gamma spectrome
try. It tracks every gamma ray randomly emitted at randomly selected location
in the sample matrix. Gamma attenuation in terms of absorption and scattering
is included in the sample matrix of a given geometrical shape, as well as attenua
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Journal of Applied Mathematics and Physics
tion in the Ge detector endcap and Ge crystal dead layer. Finally, gamma ab
sorption in the Ge crystal leads to events recorded in the gamma peak, quanti
fied as peak efficiency of the detector. Gamma scattering with partial escape
from the Ge crystal leads to events outside of gamma peak in the gamma spec
trum, contributing to the total efficiency of the detector. Also, gamma scattering
from the lead shielding (not shown in Figure 1) is included. The Gespecor pro
gram accepts as input all materials elemental composition, dimensions, and den
sities. We performed simulations with 106 gammaemission events in each stu
died case. Therefore, the calculations are realistic representations of the labora
tory measurement systems. The density corrections are built into the calcula
tions, whereas the coincidencesumming corrections [46] [48] can be included
but are negligible for 137Cs.
In the initial step, we calculated peak and total efficiencies assuming homoge
neous samples, referred to as bulk efficiencies, or Bulk efficiency model. They
Figure 1. Two counting geometries which are subjects of Monte Carlo simulations: 0.6L
and 1L cylindrical containers, and 134% and 48% Ge detectors. The relative dimensions
are to scale. The actual dimensions are given in Table 1.
Table 1. Dimensions of containers and Ge crystals. Peak and total efficiencies for bulk sample as well as max, min, and average
positions of hot particle.
Sample container
Ge crystal
Source
position
Detection efficiency
Ratio max/min
Deviation of average
from bulk (%)
Radius (cm)
Height (cm)
Volume (L)
Radius (cm)
Height (cm)
Relative efficiency (%)
Peak
Total
p
/
t
Peak
Total
p
/
t
Peak
Total
p
/
t
5.45
3.10
max
5.614E−02
1.931E−01
2.907E−01
73.6
−0.44
10.72
5.95
min
7.623E−04
5.501E−03
1.386E−01
35.1
−0.45
1.0
48
bulk
6.607E−03
3.372E−02
1.959E−01
2.1
0.01
average
6.578E−03
3.357E−02
1.960E−01
5.56
4.33
max
9.870E−02
2.731E−01
3.613E−01
18.5
−1.02
6.19
8.80
min
5.334E−03
2.404E−02
2.219E−01
11.4
−1.13
0.6
134
bulk
2.185E−02
8.221E−02
2.658E−01
1.6
0.11
average
2.163E−02
8.129E−02
2.661E−01
L. T. Chu et al.
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Journal of Applied Mathematics and Physics
are given in Table 1. We also calculated efficiencies for hypothetical hot particles
positioned at the closest and farthest locations, designated as max and min in
Figure 1, respectively. It is seen that the bulk efficiency values are between those
for the min and max positions. The ratio of peak efficiencies for the max and
min positions is 73.6 for 1 L and 18.5 for 0.6 L. This can create significant bias in
determination of hotparticle activity.
Let us abbreviate counting rate in the peak as
R
. The “true” activity deter
mined using particle peak efficiency
p
is equal to
A Rp=
. The activity deter
mined using bulk peak efficiency
pb
is equal to
bb
A Rp
=
. The Bias is defined
as
( )
( ) ()
Bias % 100 1 100 1 .
bb
A A pp= −= −
(1)
It is seen that the Bias does not depend on the activity, only on the efficiencies.
Therefore, this investigation focuses on the efficiencies only. The discussion
about hot particles having different activities is deferred to Section 5.
To study the effects associated with hot particles, we calculated the peak and
total efficiencies of 2048 individual particles randomly distributed in either 0.6L
or 1L containers, one particle at a time. The random positions of particles were
calculated first using the algorithm for cylindrical coordinates [40]. In this algo
rithm, the height of particle position is proportional to a random number, while
the radius of particle position is proportional to the square root of a random
number. The azimuth angle is not important in this case because of a cylindrical
symmetry. Then, the 2048 random particle positions were supplied to the Ges
pecor program, which calculated peak and total efficiencies at these positions.
To verify the randomness of particle positions, we hypothesize that the average
efficiency for all particles should approximate that of the bulk efficiency calcu
lated above. The average peak and total efficiencies for all 2048 particles are also
listed in Table 1. The deviations of average efficiencies from the bulk efficiencies
are about −0.5% for 1L container and about −1% for 0.6L container, whereas
the deviations for the
p
/
t
ratios are substantially smaller. The randomness of
particle positions is judged satisfactory for the purpose of this study.
The biases were calculated using Eq. 1 for the 2048 particles and are plotted as
histogram in Figure 2 for 1L container. They range from about −100% to 700%,
or by a factor of 8. This is less than 73.6 listed in Table 1, however, it is statisti
cally unlikely to have a particle located at either a min or a max position. The
frequency distribution of the biases is depicted in Figure 3 (distribution a, green
points). This distribution resembles an exponential, reflecting exponential at
tenuation of gamma radiation in the sample.
3. OneParticle Model
The 1particle model assumes presence of a single hot particle in the sample.
This model provides a foundation for the
n
particle model to be described in the
next section. Let us consider a particle in the sample matrix located at a distance
r
from the Ge detector. In a simplified picture, we neglect all possible angles and
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Journal of Applied Mathematics and Physics
Figure 2. Histogram of Bias from single particle efficiency when using bulk efficiency for
1L container.
Figure 3. Frequency plots for various Bias distributions for 1L container. (a) Bias bulk
from single particle, bin size 25%; (b) Bias for 1particle model, bin size 5%; (c) Bias for
n
particle model applied to 1 particle, bin size 10%.
finite sizes of both sample and the detector. For a gamma photon to be detected
in the peak with efficiency
pr
, it must be attenuated in the sample and detected
with an intrinsic peak efficiency of the detector
pd
. We thus have
( )
0
exp .
rd
pp r
µ
= −
(2)
Similar considerations apply to the total efficiency
tr
and intrinsic detector
td
.
However, the total efficiency is enhanced by the scattered radiation in the sam
ple originating from gamma photons emitted at the particle location
r
. There
fore,
( ) ( )
0
exp exp .
rd s
tt r r
µµ
= −
(3)
Also,
0
minor terms,
as
µµµ
=++
(4)
where
0
µ
is a total gamma attenuation coefficient,
a
µ
is a gamma absorption
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Journal of Applied Mathematics and Physics
coefficient, and
s
µ
is an incoherent (Compton) scattering coefficient [49].
By manipulation of Equations (2) and (3), we obtain
0
.
s
d
rr
d dr
t
pp
p pt
µµ
=
(5)
It follows that
p
should be a power function of
p
/
t
,
( )
,
h
p g pt=
(6)
where
g
and
h
are coefficients. In this way
p
/
t
carries some information about
hotparticle peak efficiency and thus its position.
The
p
values are plotted as a function of
p
/
t
for the 1L container in Figure 4.
The densely located 2048 points lump into a gray area. The reason for such wide
distribution is that it involves random particle locations and all directions of
gamma emissions from a particle, as well as taking the ratio of two random va
riables
p
/
t
. Then an unconstrained powerlaw fit was made to the points ac
cording to Equation (6) resulting in
pu
, depicted by the green curve in Figure 4
representing the most probable
p
.
The sequence of analyzing the data is as follows. For each simulated particle at
its location, one calculates
p
/
t
obtained from the MC simulation. Experimental
ly, for a single radionuclide, it would correspond to taking a ratio of the counts
in the peak to the total counts in the gamma spectrum. Then, one reads the most
probable value
pu
from the green curve in Figure 4. Subsequently, one calculates
the Bias from Equation (1), by substituting
pu
for
pb
.
The histogram of Bias values for 1particle model is depicted in Figure 5. It is
seen that bias now spans the range between about −60% and 60%, and it is sig
nificantly reduced from that for the bulk efficiency in Figure 2. The frequency
distribution for 1particle model is depicted as curve b in Figure 3 (blue points).
Figure 4. Plots of
p
as a function of
p
/
t
for the MC data and various fits described in the
text.
L. T. Chu et al.
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Journal of Applied Mathematics and Physics
Figure 5. Histogram of Bias from single particle efficiency when using 1particle model
efficiency for 1L container.
It is a narrow symmetric distribution reflecting significant reduction in the dis
persion of the results.
4. nParticle Model
The
n
particle model assumes that there are 1 or more hot particles in the sam
ple, all having the same activity. It can be realized by simple grouping of the
2048 MC points. For instance, one can group 2 particles (1024 cases), 4 particles
(512 cases), etc., until finally arriving at 1 case of 2048 particles. The average va
riables for
n
independent particles are given by
1
,
1
ni
i
n
pp n
=
=∑
(7)
1,
1ni
i
ntt n=
=∑
(8)
,
n nn
pt p t=
(9)
The reason for such definitions of averages is that Equation (9) is the one that
would be realized experimentally. The averages are “true” values as calculated by
the MC program, and the Bias when using Bulk efficiency model is generalized
from Equation (1):
( )
( )
Bias % 100 1 .
b
n
pp= −
(10)
The values of
2048
p
,
2048
t
, and
2048
pt
are listed in Table 1 under
source position in the average row (both geometries). As discussed in Section 2,
the averages for 2048 particles approximate bulk efficiency calculations
pb
,
tb
,
and
( )
b
pt
very well. This average for 1 L container is depicted as an open cir
cle in Figure 4, which does not lay on the green
pu
curve. The reason is that
n
pt
(Equation (9)) is not equal to
1
1
nii
i
tn p
=
∑
.
We performed another powerlaw fit to the data in Figure 4, this time we
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constrained the fit by passing through the
2048
pt
point, resulting in an
orange
pc
curve. It follows that, regardless of the number of hot particles present,
the most probable measured
p
/
t
will fall between these two curves. Therefore, we
need to interpolate between them in such a way that if
p
/
t
is close to
2048
pt
, it
will be weighted towards
pc
; if it is far from
2048
pt
, then it will be weighted
towards
pu
. We tried linear and quadratic interpolations without success because
the interpolation has to be steep in the vicinity of
2048
pt
.
A satisfactory interpolation can be obtained however, by application of the
SDT [50]. SDT is concerned with distinguishing between the signal and noise.
We assume Gaussian distributions for both signal and noise, with the mean and
sigma given as
{ }
signal signal
,
µσ
and
{}
noise noise
,
µσ
, respectively. We designate a
Gaussian (normal) distribution function (an integral of the Gaussian probability
density function from minus infinity to the Deviate) with mean and sigma, and
its inverse as
{ }
( )
Probability Deviate, mean,sigma ,= Φ
(11)
{ }
( )
1
Deviate Probability, mean,sigma .
−
= Φ
(12)
For the set of 2048 points, we find the minimum and maximum values of
p
/
t
,
( )
min
pt
and
( )
max
pt
, respectively. For any average of
n
particles, the SDT
probability of signal rejection,
Pr
can be written as
( )
( )
( )
( )
min 2048
min
2048
max 2048
max 2048
, , (13a )
, . (13b)
nn
r
nn
pt pt pt pt
pt pt
Ppt pt pt pt
pt pt
− ≤
−
=−
>
−
Then, by combining Equations (11)(13), we obtain the SDT probability of
misses
Pm
and hits
Ph
as
{ }
( )
{ }
1,, , , ,
m r noise noise signal signal
PP
µσ µ σ
−
=ΦΦ
(14a)
.
1
hm
PP
= −
(14b)
We determined that a mean and sigma of {0, 1} for the noise, and {0.6, 0.5} for
the signal, provided sufficient convergence.
Finally, the interpolated value of
pinterp
between the unconstrained fit
pu
and
the constrained fit
pc
from Figure 4 (green and orange curves, respectively) is
given by
.
interp h u m c
p Pp Pp= +
(15)
The Bias for the
n
particle model is given by
( )
()
Bias % 100 1 .
interp
n
pp= −
(16)
The performance of various models is listed in Table 2 for 1L container. The
quantity of interest is the Bias(%) from “true” efficiencies known from the MC
calculation, its minimum (Min) and maximum (Max) values, as well as standard
deviation (Std Dev). Table 2 shows that for 1 particle, the Bulk efficiency model
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Journal of Applied Mathematics and Physics
resulted in a significant minimum and maximum bias and bias standard devia
tion. This was already seen in Figure 1 and Figure 2. As the number of particles
increases, the bias and all its measures decrease since the inhomogeneity drops.
Application of 1particle model to 1 particle present significantly reduces the bi
as and its standard deviation. Application of
n
particle model reduces all meas
ures of bias as compared with the Bulk model, for all particles studied. The
n
particle model applied to 1 particle performs slightly worse than the 1particle
model. This is seen in the bias measures in Table 2, as well as in the frequency
distribution with right tail (curve c in Figure 3).
The results of the Bias for the 0.6L container are listed in Table 3. They exhi
bit the same trends as for the 1L container, however, of smaller magnitude
Table 2. Performance of the Bulk,
n
particle, and 1particle models expressed as bias
from known efficiency for several number of particles and 1L container.
Model
Bias (%)
Number of
particles
1
2
4
8
16
32
64
Bulk
Min
−88.1
−86.2
−79.1
−76.2
−54.9
−48.2
−29.6
Max
701.2
430.1
257.9
151.0
84.4
41.2
33.6
Std Dev
114.7
79.3
58.1
41.4
29.7
20.8
14.9
n
particle
Min
−61.7
−63.3
−56.9
−56.9
−35.4
−22.7
−18.9
Max
137.5
117.1
74.0
62.5
43.3
21.2
24.4
Std Dev
35.7
27.8
22.5
20.4
15.9
11.5
9.6
1particle
Min
−61.7
Max
57.9
Std Dev
26.5
Table 3. Performance of the Bulk,
n
particle, and 1particle models expressed as bias
from known efficiency for several number of particles and 0.6L container.
Model
Bias (%)
Number of
particles
1
2
4
8
16
32
64
Bulk
Min
−74.7
−73.3
−66.9
−49.8
−41.6
−22.0
−13.7
Max
326.9
223.4
157.7
97.3
55.4
27.3
14.1
Std Dev
70.4
50.6
36.2
26.3
17.3
10.3
7.2
n
particle
Min
−49.1
−45.7
−42.1
−32.9
−20.7
−12.4
−6.5
Max
74.1
60.5
50.8
39.5
17.5
11.3
8.0
Std Dev
25.4
19.2
14.7
11.1
7.8
6.0
3.7
1particle
Min
−49.2
Max
52.6
Std Dev
23.2
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Journal of Applied Mathematics and Physics
reflecting relatively less inhomogeneity in this smaller container.
In order to compare model performances for the two containers, we define
Bias Range (%) as the difference between the Max and Min bias. Then, the im
provement factors for the Bias Range and Bias Std Dev (%) are calculated as the
ratios of the corresponding values from the Bulk model to the ones for the
1particle and
n
particle models. The improvement factors are depicted in Fig
ure 6 for 1 and more particles present in the sample. It is seen that there are sig
nificant improvements in Bias Range and Bias Std Dev when using the calibra
tion models developed here. The highest improvements by a factor ranging from
about 3 to 6.5 are when using 1particle model. The same case of 1 particle
present, when applying the
n
particle model to it, resulted in improvement fac
tors from about 3 to 4. As the number of particles increases, the improvement
factors drop to between 1.5 and 2 for 64 particles. For small number of particles
(1 to 4), the improvement factors are higher for 1L container than for the 0.6L
container; this trend is reversed for larger number of particles in most cases.
5. NonEqual Particles and MultiPeak Radionuclide
The
n
particle model assumes that all particles have the same activities. It was
observed that hot particles from nuclear accidents exhibit distribution of sizes
[14], often assumed lognormal [15]. Then, constant specific activity implies
lognormal distribution of activities as well. This situation results in even more
inhomogeneity than for equal particles, because a few of very hot particles do
minate activity of the sample.
Nonequal particles can be easily simulated within the present data set by po
sitioning several equal particles in the same location. Since the derived
n
particle
Figure 6. Improvement factors for Bias Range and Bias Std Dev when using 1particle
and
n
particle models compared with the Bulk model, for 1L and 0.6L containers.
L. T. Chu et al.
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Journal of Applied Mathematics and Physics
model interpolates the efficiencies between 1 particle and the bulk sample, it
should be independent of a specific assumption about the number or location of
the particles. Therefore, it should apply to nonequal particles as well.
To test this hypothesis, we created two cases of nonequal particles, for 1L
container. In the first case we have 2 particles, one is assumed twice as radioac
tive as the other. Therefore, the average peak efficiency from Equation (7) is
( )
12
2
13 2p pp= +
. We have used 2 equal particles from the MC set of 2048
and created a group of 2 nonequal particles. We have 1024 such groups to per
form statistics on. For the second case, we repeated 1st particle 4 times, 2nd par
ticle 2 times, and took the 3rd and 4th particles as is. The average peak efficiency is
( )
1 234
4
18 4 2p p ppp= + ++
. We thus have 512 such groups. Similarly, we
calculated
n
t
and
n
pt
,
2,4n=
. Then, we apply the
n
particle model by
calculating
interp
p
from Equation (15) and study the Bias from Equation (16).
The results are given in Table 4 in terms of Bias Range and Bias Std Dev. Al
so, the results for equal 1, 2, and 4 particles are reproduced from Table 2. It is
seen that the values for 2 nonequal particles are laying between those for 1 and
2 equal particles. The reason is that one particle dominates by assumption. For
the case of 4 nonequal particles, the values are between those for 2 and 4 equal
particles because of assumed distribution of activity among particles and domi
nation by the hotter ones. Nevertheless, the
n
particle model performed well and
the calculated values of Bias Range and Bias Std Dev are significantly improved
from those of the Bulk model.
In this work we have considered a single gamma peak of 661.11 keV from
137Cs. The
n
particle model based on
p
/
t
can be extended to multipeak radio
nuclide as follows. Let
A
represent the radionuclide activity of a hot particle. The
counting rate in gamma peak
j
is given by
,,
,
p j eff j
R Ap
=
(17)
where
,eff j
p
is an effective peak efficiency, which includes gamma intensity, den
sity correction in the sample matrix and any coincidencesumming correction. For
k
peaks of the radionuclide in the gamma spectrum, the total counting rate in all
peaks is equal to
,,
11
.
p p j eff j eff
j
k
j
k
R R A p Ap
= =
= = =
∑∑
(18)
Table 4. Performance of the
n
particle model for nonequal particles in 1L container.
Number of particles
Equal particles
1
2
4
Bias Range (%)
199.2
180.3
130.9
Bias Std Dev (%)
35.7
27.8
22.5
Nonequal particles
2
4
Bias Range (%)
188.0
168.0
Bias Std Dev (%)
29.6
26.0
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Journal of Applied Mathematics and Physics
Similarly, for the total counting rate,
,,
11
.
t t j eff j eff
j
k
j
k
R R A t At
= =
= = =
∑∑
(19)
Therefore, the observed peaktototal ratio for a multipeak radionuclide is
given by
( )
.
p t eff eff
eff
pt R R p t= =
(20)
The lefthand side of Equation (20) can be measured, while the righthand
side calculated by the MC simulation. The
n
particle model can be used as earli
er in Sections 4, 5, with
p
and
t
replaced by
peff
and
teff
.
6. Discussion
A overview of the origins and behavior of hot particles was provided in Section
1. Such particles are normally encountered in environmental samples following
nuclear accidents. Digital radiography is typically used for samples spread on a
surface, while digital tomography is not always practical for measuring many
environmental samples. Surveillance samples for gamma spectrometry are typi
cally large to increase sensitivity and to provide better sampling of radioactive
contamination. However, voluminous samples may be inhomogeneous due to the
presence of hot particles. Applying the Bulk efficiency model leads to significant
bias of measured activity. One approach to this problem is repetitive mixing and
measuring of a sample (from 25 to 100 times [36] [38]). While this method can
provide relatively accurate average of the measured activity, is not very practical
for many samples, and the dispersion evaluated by means of a standard devia
tion can be large and increasing as the number of hot particle decreases.
In this investigation we provided a method to reduce this dispersion. Our goal
was to study the behavior of hot particles randomly distributed in the soil ma
trix, for two counting geometries: 1L and 0.6L cylindrical containers. Using
MC simulation, we calculated the peak and total efficiencies of 2048 individual
hot particles at random locations in each container. We focused on a single ra
dionuclide of 137Cs and its 661.66keV gamma ray. The presence of counts in the
gamma peak in the spectrum represents gamma transmission, whereas anywhere
in the gamma spectrum, both transmission and scattering. On this basis, we de
rived a powerlaw relationship between peak efficiency and peaktototal ratio.
This relationship was confirmed by the MC simulation. Since peaktototal ratio
is a measurable parameter, the powerlaw relationship can provide a more accu
rate value of the efficiency than the bulk efficiency. We called this approach
1particle model. It was shown that the 1particle model reduced the Bias Range
6.5 times and Bias Std Dev 4 times compared to the Bulk efficiency model for
1L container, and 4 and 3 times for 0.6L container, respectively.
Subsequently, identical individual particles were combined into
n
particle
groups, which resulted in the
n
particle model. The averages for peak and total
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Journal of Applied Mathematics and Physics
efficiencies were calculated as well as their peaktototal ratio. From the pow
erlaw relationship, the most probable value of peak efficiency could be obtained
from the empirically available peaktototal ratio. The complication arose from
the fact that the most probable peak efficiency lays between the singleparticle
powerlaw and the powerlaw constrained by the bulk efficiency point. Fortu
nately, an innovative interpolation was developed, based on a formulation from
the Signal Detection Theory, between the two powerlaw curves. The
n
particle
model works well for any number of particles. When applied to a singleparticle
case, the reduction in Bias Range and Bias Std Dev was between 3 and 4. These
are improvement factors when using this model compared with the Bulk model.
As the number of hot particles increases, the inhomogeneity decreases, and im
provement factors decrease to between 1.5 and 2 for 64 hot particles. We also
simulated groups of nonequal hot particles and found that the
n
particle model
is independent of the number, location, and size distribution of hot particles.
If this approach is applied to repetitive mixing and measurement method for
inhomogeneous sample, the accuracy of the average is expected to have low bias
as before, however, the standard deviation will decrease several times. If only one
measurement is made on an unknown sample, this method guarantees reduction
of the bias several times for inhomogeneous samples without the need of know
ing the details of inhomogeneity.
7. Conclusion
We developed a novel calibration of a gammaray spectrometer using the rela
tionship
( )
h
p g pt=
, where
g
and
h
are coefficients and
p
/
t
is a ratio of
peaktototal efficiencies. This method can be used to reduce the variance of
measured activity in bulk environmental or food samples containing hot par
ticles as compared with the homogenous calibration. So far, we were able to ap
ply it to a singlepeak singleradionuclide, such as 137Cs, which is known as the
most important gamma emitter for aged fallout from nuclearpower accidents.
We also derived equations to accommodate multiplepeak radionuclide. At this
time, the method has not been shown suitable for characterizing samples con
taining mixtures of radionuclides.
Acknowledgements
This work was partially supported by the US FDA FERN Cooperative Agree
ment 1U18FD005514.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this
paper.
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