A Simplified Gamma-Ray Self-Attenuation
Correction in Bulk Samples
A. E. M. Khater,1,2* Y. Y. Ebaid,3
1
2
National Centre for Nuclear Safety and Radiation Control, Atomic Energy Authority, Egypt
Physics Department, College of Sciences, King Saud University, Riyadh, Kingdom of Saudi Arabia
3
Physics Department, Faculty of Science, Fayum University, Fayum, Egypt
Abstract
Gamma-ray spectrometry is a very powerful tool for radioactivity measurements.
For accurate gamma-ray spectrometry, certain correction measures should be considered,
for instance, uncertainty in the photo-peak efficiency due to the differences between the
matrix (density and chemical composition) of the reference and the other bulk samples.
Therefore, gamma-ray attenuation correction factors are of major concern for precise
gamma-ray spectrometry. Simple practical correction for the photo-peak efficiency, due
to discrepancies in both the samples matrices and densities (self-attenuation), is
performed in this study. This study suggests a brief measurement of relative photons
transmission through both reference and unknown bulk samples where the variations of
photon transmissions are linearly correlated to the samples’ densities. Specific correction
factors would be produced for each analyzed sample to be considered when their
*
Corresponding author present address; P.O. Box 2455, 11451 Riyahd, Kingdom of Saudi Arabia.
Email: khater_ashraf@yahoo.com (A. Khater); yebaid@yahoo.com (Y. Ebaid)
1
activities are calculated. Practically, the suggested method was verified and succeeded in
improving the obtained results.
Keywords: Gamma-ray spectrometry; attenuation correction; bulk sample
Introduction:
Gamma-ray spectrometry using hyper pure germanium (HpGe) detectors has been
an essential and principal spectroscopy technique in almost all radioactivity
measurements laboratories worldwide. Its major advantages are being non-destructive,
multi-elements analysis, simplified regarding sample preparation, i.e. mostly no need for
any chemical separation processes, and its applicability for all types of samples, etc. Low
energy gamma emitters are used in many scientific studies in a wide range of
applications. For example, 210Pb (46.5 keV) is used for determining sedimentation rate in
lakes, estuaries and the coastal marine environment during the last century. Also it is used
to study the atmospheric fluxes and mixing in the troposphere. Additionally
234
Th (63.3,
92.4 and 92.8 keV) is extensively used as a natural tracer to study biological mixing and
particle scavenging processes in recent sediment (San Miguela et al., 2004).
However, the precise determination of the activity concentration of each
radionuclide requires the determination of full energy efficiency calibration for a given
geometry. Therefore, a detection efficiency curve, known as efficiency calibration, over
the energy region of interest must be established precisely in advance. The detection
efficiency at certain gamma-ray energy and sample geometry is given by;
 ( E , n) 
C ( E , n)
t. f ( E , n). A
2
Where:
C(E,n): net photo-peak count of gamma-ray transition with energy E of radionuclide n,
t:
counting time, s,
f(E,n): branching ratio, number of photon with energy E per hundred disintegration of
radionuclide n,
A:
activity concentration in Bq of radionuclide n.
The detection efficiency curve depends not only on a detection system but also on
both the sample shape and matrix (Saegusa et al., 2004). Efficiency calibration can be
performed theoretically using Monte Carlo computation techniques or semi-empirically
using analytical method. Experimentally, the detection efficiency curve can be performed
using standard samples that contain a set of radionuclides with known activities and
cover the gamma-ray energy range of interest (usually from 35 to 2000 keV). Standard or
reference samples should have the closest specifications, regarding geometry and
matrices (apparently density and composition), to the analyzed samples (Abbas et al.,
2006; Gurriaran etal, 2004; Vargas et al., 2006). The accurate determination of the
photopeak efficiency curve for a given sample matrix represent the main challenge in
gamma-ray spectrometry (Quindos, 2006). Practically, the samples geometry (shape and
sample-detector geometry) can be easily reproduced. However, the major source of
possible error in efficiency calibration remains due to the difficulties to reproduce the
same matrix and chemical composition of the standard samples as that of the other bulk
samples that could have vast verities in both densities and compositions.
3
An important correction applied in gamma-ray spectrometry of bulk samples is
the correction for photon attenuation within the source material itself that known as selfattenuation or absorption correction (Boshkova, 2003). For a given geometric setup, the
correction factor is expressed as the ratio of efficiency of standard to that of the sample:
CS ( E ) 
 ( E , S tan dard )
 ( E , sample)
In practical applications, standard and sample materials might be entirely
different. Environmental samples may greatly differ in their chemical composition, even
of the same matrices (such as soil and sediment samples), and their density ranging from
extremely low densities up to 2.0 g/cm3. Self-absorption correction can be determined
experimentally or using the Monte Carlo computation techniques or using analytical
methods.
The procedure for self-absorption factor determination includes firstly calculation
of CS that is obtained for various densities ρ and photon energies E. Then data collected
for each photon energy are fitted to an appropriate function Cs(ρ) or Cs(E, ρ) that are
written as
Cs(ρ)
= a exp(-b ρ)
or
Cs(E, ρ) = a(E) exp[-b(E) ρ]
Where a and b are the adjustable parameters.
4
For low energy gamma-ray (below about 100 keV) these formulae are applicable
only for materials of similar composition on account of the relation between the mass
attenuation coefficient and the atomic number (Jodlowski, 2006). Gamma-ray photons
are known to be attenuated through the material according to the following relation;
I  I o .e(  m . x ) &
I  I o .e ( m .x.  )
Where:
Io:
the photons, with energy E, intensity without attenuation
I:
the photons, with energy E, intensity after attenuation
μ:
the linear attenuation coefficient, cm-1,
μm:
the mass attenuation coefficient, μm = μ/ ρ, cm-2/g.
x:
the sample thickness (or effective thickness)
Jodlowski (2006) compared the different methods for self-absorption correction in
gamma-ray spectrometry of environmental samples and conclude the following
(Jodlowski, 2006):
-
The experimental method is time consuming and inconvenient. It requires that the
curves be fitted to a small number of measurement data, which gives rise to a
relatively high uncertainty.
-
Monte Carlo techniques are not so widely adopted in the laboratory conditions, as
they require considerable skill and experience in computer simulations.
5
-
The exact analytical description of self-absorption is a complex task that is why
simplified models are adopted instead. In the first place, it is assumed that the selfabsorption correction is proportional to the term exp (-μ x) or exp (-μm ρ x). Another
widely applied analytical formula providing a simplified description of selfabsorption in cylindrical samples is involving the integration of photons of the
specified energy coming from subsequent sample layers and reaching the detector.
Self-absorption correction, C sa , for the sample with reference to air (matrix μ ≈ 0) is
given by the following equation:
C sa 
 .x
1  e  x
Another analytical method was proposed where in order to apply Cs computing
techniques, it is required that attenuation coefficients of the standard and sample
materials be known before hand. For high energy range (more than 100 or 200 keV) of
gamma-ray photons, the mass attenuation coefficient is obtained on the basis of the
sample approximate chemical composition, assuming that for a given energy value the
value of Cs depends on the sample density exclusively. For low energies (below 100
keV), the mass attenuation coefficient in different materials may vary significantly and
the attenuation coefficient has to be determined experimentally.
The transmission method developed by Cutshall et al (1983) where a point-like
radioactive source is positioned above the sample located on the detector and the number
of counts in the full energy peak is measured. The sample self-absorption correction
factor Cs with the reference to the standard is given as:
6
Cs 
ln(I c / I s )
1  (I c / I s )
Where Ic, Is are the transmission experiment results (detector count rates) for the
standard and sample, respectively (Cutshall et al., 1983).
The aim of our work is to shed more light on the significant of self-absorption correction
for precise gamma-ray spectrometry of bulk samples and a simple practical procedure to
correct it.
Experimental work
Four point gamma-ray emitter sources (241Am,
133
Ba,
137
Cs, and
60
Co) with the
following energy transitions (59.5 keV, 81.2 keV, 276 keV, 302.4 keV, 356 keV, 383.9
keV, 661.6 keV, 1173.2 keV and 1332.5 keV) were utilized to perform transmission
measurement for standard and other samples. Each sample was placed on the top of the
detector, and then the four point sources were put above the sample. Enough distance was
left between the detector and the point sources to maintain acceptable dead time. To
obtain acceptable low statistical error, counting time was 2000 second
- Samples preparation
Ten samples of different matrices (water, grass, soil and geological ore), with the
same geometry, have been prepared to test the gamma-ray attenuation in relation to
samples matrix and density differences. Samples were prepared in identical
polypropylene containers and tightly closed. This set of samples cover a wide range of
7
appeared densities of 0.48 – 1.69 g/cm3. Another set of samples was prepared to have the
same geometry and matrix, but different appeared densities that ranged from 0.98 to 1.42
g/cm3. Another set of samples was prepared to have the same geometry and appeared
density (- 5%), and of different matrices (water, soil and geological ore).
- Gamma ray spectrometry:
Gamma-ray spectrometer with a GX4019-7500Sl CANBERRA extended range
electrode germanium detector with a CANBERRA model 2002CSL preamplifier was
used. The hyper pure germanium (HpGe) detector has a relative efficiency of 40% and
full width at half maximum (FWHM) of 1.9 keV for
60
Co gamma energy transition at
1332.5 keV. It is connected to 8k computerized multi-channels analyzer where gamma
spectrum analysis performed using Genie-2000 gamma-ray spectroscopy software by
Canberra.
Results and Discussions:
Gamma-ray spectrometry based on hyper pure germanium detectors is a very
powerful tool that has a very wide range of applications in radiation measurement,
generally, and specially in environmental radioactivity measurement in different bulk
samples. Many studies have been focused on the accurate detection efficiency calibration
using different, experimental, Monte Carlo simulation and analytical, techniques. Other
studies have been concerned with background reduction, by passive and/or active
shielding, for low level measurement; quality control, optimum sample-detector
geometry arrangement, and other aspects. However, the variations of chemical and
8
physical properties of the bulk samples could be the main source of noticeable
uncertainty, especially for the samples with high apparent densities and different
chemical compositions from that of the standard samples used for efficiency calibration
and specially in relatively low gamma-ray transmission energies (below 100-200 keV). In
the low gamma-ray energy ranges, correction of self- attenuation (absorption) of gammaray in the bulk samples is essential for accurate results. Different methods have been
suggested and applied for self-attenuation corrections. It is using experimental, Monte
Carlo simulation and analytical techniques (Sigh et al., 2004; Pilleyre et al., 2006; Sima
and Dovlete, 1997; Cincu, 1992; Korun, 2000). All this techniques can overcome the
self-attenuation problem in many cases but can not ensure it, especially in low energy
range and for some samples of high densities that far from the densities of the calibration
sources. In this work we are suggesting a modified practical method for attenuation
correction based on the relative gamma-ray transmissions (I/Io) through both standard and
the different bulk samples relatively to the air or water samples of the sample geometry.
The relative gamma-ray transmissions for a set of bulk samples of different
densities (0.48-1.69 g/cm3) and different matrices for different photon energies (59.51332.5 keV) are given in Table 1 and shown in Figure 1. The variation of the relative
transmission is very wide especially for low energies (59.5 and 81.2 keV) that could be
because of the density and matrix differences. These variations are clearly reduced at the
high energies range of gamma rays.
Another set of samples, of the same matrix and different densities (0.98-1.42
g/cm3) have been examined. The relative gamma-ray transmission (I/Io) for energy range
59.5-1332.5 keV and for different sample densities are given in Table 2 and shown in
9
Figures 2 and 3. Both figures show the variation of the relative gamma ray transmission
as a function of energy and sample density where the variation of the relative
transmission is linearly correlated to the sample density and polynomially (exponentially
growth) correlated to the photon energy.
The last set of samples has the same apparent density and different matrices (i.e.
chemical composition). The relative gamma-ray transmission as a function of gamma-ray
energy are given in Table 3 and shown in Figure 4. It is obviously clear that the variations
in the transmission of gamma-rays of different energies, due to the variation in the
chemical composition of the samples, are considerable. This figure show the variation in
the gamma-ray self attenuation either due to the density or the chemical composition or
both of them. The uncertainty in the gamma-ray spectrometry, due to the samples’
densities or chemical compositions or both, has different patterns. Unless the difference
in density and chemical composition is fairly close to that of the reference, the selfattenuation correction of the bulk sample is a necessity to eliminate its effect on the
accuracy of the detection efficiency. In the following section of this paper, we shall
explain our suggested method for self-attenuation correction without a tedious work of
the experimental approach or the need to mathematical skills or the simplified assumption
of both Monte Carlo simulation and analytical approaches.
In this work, we are suggesting a simple method to overcome the variation in the
counting efficiency due to self-attenuation in the bulk samples. This method is based on
measuring the relative transmission of gamma-ray of different energies as a ratio of gamma-ray
10
photo-peak count rate pass through the reference standard and other (unknown) samples to that
through the air or other samples such as water of the same geometry, (I/I0)std and (I/I0)unk,
respectively. To overcome the errors due to sample’s radioactivity contributions to the point
source counts, point sources were selected so that they have relatively higher activity than that of
the samples themselves. Also, they were counted together for relatively short time. The
advantage of this method is that it takes into consideration both the density and matrix effect
while correcting for the counts, i.e. detection efficiency. Accordingly, the activity concentration
would be corrected using the following formula
Aunk 
CR unk  (I
)
I 0 std
  f  (I I )unk
0
(4)
Where
CRunk : The unknown sample’s count rate at the specified photo-peak energy

: The photopeak efficiency for certain geometry

: The photopeak intensity
(I/I0)std : relative transmission of gamma-ray through the standard reference material.
(I/I0)unk : relative transmission of gamma-ray through the unknown measured sample.
Equation no. 4 could be improved to the following formula;
Aunk 
CRunk

f
(5)
Where  is the correction factor calculated by dividing the point source counts passing
through the standard reference material by that of unknown sample according to the following
equation:
11

(I
(I
I0
)Std
I0
)Unk
(6)
Application: The IAEA-RGU-1 (uranium ore) reference sample was used to perform the
efficiency calibration for the energy transition of 63.3 keV (3.6 %) of the
234
Th. The
234
Th (in
equilibrium with its parent 238U) is usually used for the specific activity assessment of 238U.
The estimated correction factors were used to calculate the
238
U activities of other ore
samples, of the same geometry and of different densities, as well as uranyl nitrate solution with
known uranium concentrations. The corrections suggested in this paper were then applied for
self-attenuation correction. The results were compared and shown in Table 4. The performed
corrections were based on the transmission differences due to the nearest energy transition (59.5
keV) of the
241
Am. Therefore, when the energy transition of interest is not available as point
source, it is recommended to produce a fitted correction curve using energies as near as possible
to that needed. Using this curve, a correction factor could be easily obtained for most of required
energy transitions. It is clearly noticed that a considerable improvements has been encountered in
the measuring process, where the percentage of error is dramatically decreased.
Conclusions:
The suggested procedure introduced in this paper is an innovative, reliable and
straightforward method to overcome the errors generally produced due to the difference
in samples matrices and densities. It also minimizes the measurement errors. There is no
need for tedious experimental work or assumptions for the mathematical approaches.
This method could be adopted within the laboratories where they encounter incoming
12
wide varieties of samples for analysis. A simple calculation program could be produced
to help the unskilled technicians in the routine measurements of gamma-ray spectrometry
for bulk sample. Finally, the applicability of this method is almost unlimited as long as
the sample is homogenous.
References:
Abbas, Mohamoud I., Nafee, Sherif, Selim, Younis S., 2006. Calibration of cylindrical
detectors using a simplified theoretical approach. Appl. Radiat. Isot 64,
1057-1064.
Aguiar, Julio C., Eduardo Galiano, Jorge Fernandez, 2006. Peak efficiency calibration
for attenuation corrected cylindrical sources in gamma ray spectrometry by
the use of a point source. Appl. Radiat. Isot 64, 1643-1647.
Boshkova, T., 2003. Effective thickness of bulk samples in close measuring gamma-ray
spectrometry. Appl. Radiat. and Isot. 59, 1-4.
Cincu, Em. , 1992. A practical method for accurate measurement of radionuclide
activities in environmental samples. Nucl. Instr. and Meth. A 312, 226230.
Cutshall, N., Larsen, , I.L., Olsen, C.R., 1983. Direct analysis of Pb-210 in sediment
samples: a self-absorption corrections. Nucl. Instr. and Meth. A 206, 309312.
Gurriaran, R., Barker, E., Bouisset, P., Cagnat, X., Ferguson, C., 2004. Calibration of a
very large ultra-low background well-type Ge detector for envoironmental
sample measurements in an underground laboratory. Nucl. Instr. and Meth.
A 524, 264-272.
13
Jodlowski, Pawel, 2006. Slef-absorption correction in gamma-ray spectrometry of
environmental samples- an overview of methods and correction values
obtained for the selected geometries. Nukleonika 51 (2), S21-S25.
Korun, M. , 2000. Calulation of self-attenuation factors in gamma-ray spectrometry for
samples of arbitrary shape. Radioanal. Nucl. Chem. 244(3) 685-689.
Pilleyre, T., Sanzelle, S., Miallier, D., Fain, J., Courtine, F., 2006. Theoretical and
experimental estimation of self-attenuation correction in determination of
210
Pb by gamma-spectrometry with well Ge detector.
Radiation
Measurements 41, 323-329.
Quindós, L. S., Sainz, C., Fuente, I., Nicolás, J., Quindós, L., Arteche,J., 2006. Correction by
self-attenuation in gamma-ray spectrometry for environmental samples. Radioanal.
Nucl. Chem. 270 (2), 339-343.
Saegusa, Jun, Kawasaki, Katsuya, Mihara, Akira, Ito, Mitsuo, Yoshida, Makoto, 2004.
Determination of detection efficiency curves of HPGe detectors on
radioactivity measurement of volume samples. Appl. Radiat. Isot 61, 13831390.
San Miguela, E.G. , Perez-Morenoa, J.P., Bolivara, J.P., Garcia-Tenoriob, R., 2004. A semiempirical approach for determination of low-energy gamma-emmiters in sediment
samples with coaxial Ge-detectors. Appl. Radiat. Isot 61, 361–366.
Sima, O. , Dovlete, C., 1997. Matrix effects in the activity measurement of
environmental samples- implementation of specific corrections in Gammaray spectrometry analysis program. Appl. Radiat. Isot. 48 (1), 59-69.
Singh, Charanjeet, Singh, Tejbir, Kumar, Ashok, Mudahar, Gurmel S., 2004. Energy
and chemical composition dependence of mass attenuation coefficient of
building materials. Annals of Nuclear Energy 31, 1199-1205.
14
Vargas, M. Jurado, Fernandez Timon, A., Cornejo Diaz, N., Perez Sanchez, D., 2002.
Monte Carlo simulation of the self-absorption correction for natural samples
in gamma-ray spectrometry. Appl. Radiat. Isot 57, 893-898
15
Table 1: Gamma-ray transmissions (I/Io) for a set of samples of different environmental matrices and for different
photon energies.
Serial
Sample types
No.
Energy, keV
Density
g.cm
-3
59.5
81.2
276
302.4
356
383.9
661.6
1173
1332.5
1
Air
0.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
2
Grass
0.48
0.81
0.86
0.88
0.90
0.89
0.89
0.93
0.92
0.92
3
Soil-1
0.95
0.59
0.75
0.82
0.85
0.85
0.87
0.89
0.90
0.91
4
Ore-1-1
0.97
0.54
0.68
0.83
0.83
0.82
0.86
0.85
0.89
0.88
5
Ore-2-1
0.98
0.65
0.62
0.83
0.84
0.82
0.84
0.87
0.89
0.91
6
Water
1.00
0.68
0.74
0.79
0.83
0.80
0.84
0.86
0.89
0.90
7
Ore-1-2
1.11
0.64
0.68
0.80
0.81
0.78
0.83
0.87
0.90
0.90
8
Ore-2-2
1.25
0.61
0.64
0.80
0.79
0.75
0.82
0.85
0.89
0.88
9
Soil-2
1.37
0.49
0.60
0.77
0.75
0.78
0.81
0.82
0.87
0.87
10
Ore-2-3
1.42
0.57
0.61
0.77
0.77
0.72
0.79
0.83
0.87
0.86
11
Soil-3
1.69
0.43
0.54
0.68
0.72
0.72
0.71
0.78
0.84
0.84
16
Table 2: Gamma-ray transmissions (I/Io) for the same sample matrix (Ore-2) with different densities
(g.cm-3) and for different photon energies.
Energy, keV
Density
g/cm3
0.98
1.11
1.25
1.42
59.5
81.2
276
302.4
356
383.9
661.6
1173
1332.5
0.65
0.64
0.61
0.57
0.72
0.68
0.64
0.61
0.83
0.80
0.80
0.77
0.84
0.81
0.79
0.77
0.82
0.78
0.75
0.72
0.84
0.83
0.82
0.79
0.87
0.87
0.85
0.83
0.89
0.90
0.89
0.87
0.91
0.90
0.88
0.86
17
Table 3: Gamma-ray transmissions (I/Io) for a set of different sample matrices with the same densities, ±5%
(g.cm-3) and for different photon energies
Sample types
Ore-1-1
Soil-1
Ore-2-1
Water
Energy, keV
Density
g/cm3
0.97
0.95
0.98
1.00
59.5
0.54
0.59
0.65
0.68
81.2
0.68
0.75
0.62
0.74
276
0.83
0.82
0.83
0.79
302.4
0.83
0.85
0.84
0.83
356
0.82
0.85
0.82
0.80
383.9
0.86
0.87
0.84
0.84
661.6
0.85
0.89
0.87
0.86
1173
0.89
0.90
0.89
0.89
1332.5
0.88
0.91
0.91
0.90
18
Table 4: Specific activity U-238, Bq/kg, calculated for reference samples using the traditional method (without
correction) and using present suggested corrections
Reference value
Calculated Value
(g/cm )
(Bq/Sample)
(Uncorrected)
Ore -2-
1.25
464.7
440.5
-5.2
466.9
0.5
Ore -2
1.42
528.9
486.6
-8.0
545.0
3.0
1.02
437.8
471.2
7.9
442.9
1.1
Sample Code
Uranyl Nitrate
solution
Density
3
Bias %
Calculated Value
(corrected)
Bias %
19
59.5 keV
0.9
81.0 keV
0.9
Y = 0.93 - 0.287 X
0.8
0.9
Y = 0.97 - 0.26 X
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
1.8
1.0
1.0
0.9
0.9
0.8
0.8
302.9 keV
0.7
0.6
0.8
1.0
1.2
1.4
1.6
0.6
0.8
1.0
1.2
1.4
1.6
0.9
0.8
1.0
1.2
1.4
1.6
Y = 0.99 - 0.117 X
0.6
0.9
0.9
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Y = 0.97 - 0.069 X
0.6
0.5
0.4
0.4
0.4
1.8
1332.6 keV
0.7
Y = 0.95 - 0.061 X
0.5
0.4
1.6
0.8
1173.2 keV
0.6
0.5
0.4
1.8
1.0
0.7
661.6 keV
0.7
1.4
0.4
0.6
1.0
0.8
0.8
1.2
0.5
0.4
1.0
1.0
Y = 0.97 - 0.155 x
0.6
0.4
1.8
0.8
356.01 keV
0.7
Y = 0.98 - 0.138 X
0.5
0.4
0.6
0.8
383.9 keV
0.6
0.4
0.4
1.8
0.9
0.7
0.5
Y = 0.97 - 0.149 x
1.0
Y = 0.98 - 0.155 X
0.6
276.4 keV
0.4
0.4
0.4
I/IO
1.0
1.0
1.0
0.4
0.6
0.8
1.0
Denisty, g/ cm
1.2
1.4
1.6
1.8
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
3
Fig 1: Gamma-ray transmissions (I/Io) for a set of different environmental samples of
different matrices with different densities (g.m-3)and for different photon energies
20
1.0
Ore-2 samples
Energy, keV
59.5
81.0
276.4
302.9
356.01
383.9
661.6
1173.21
1332.6
0.9
I/I0
0.8
0.7
0.6
0.9
1.0
1.1
1.2
1.3
Denisty, g/cm
1.4
1.5
1.6
3
Fig 2: Gamma-ray transmissions (I/Io) for the same sample matrix (Ore-2) with different
densities (g.m-3) and for different photon energies
21
0.95

0.90

0.90
2
±
R =0.97
2
R =0.97
0.85
0.85
0.80
0.80
0.75
0.75
0.70
Polynomial regression equation:
2
3
4
Y = A + B1*X + B2*X + B3*X + B4*X
A = 0.535
B2= -3.08397E-6
B4= -6.32973E-13
0.65
B1= 0.00169
B3= 2.36547E-9
0.70
Polynomial regression equation:
2
3
Y = A + B X + B X +B3X
1
2
0.65
A = 0.60664
I / I0
B1 = 8.57177E-4
B2 = -8.82923E-7 B3 = 3.06021E-10
0.60
0.60
0
200
400
600
800
1000
1200
1400
0
200
400
600
800
1000
1200
1400
0.90
0.90


0.85
0.85
2
R =0.94
2
0.80
R =0.94
0.80
0.75
0.75
0.70
Polynomial Regression Equation:
0.70
2
3
Y=A+B X+B X +B X
1
2
3
0.65
Polynomial Regression Equation:
0.65
2
3
Y=A+B X+B X +B X
1
2
3
0.60
0
200
400
600
800
0.60
A = 0.56784
B1= 9.60098E-4
B2= -1.01224E-6
B3= 3.54804E-10
1000
1200
0.55
1400
0
200
400
600
800
A =
0.53003
B1 =
0.001
B2 =
-1.04263E-6
B3 =
3.61575E-10
1000
1200
1400
Energy, keV
Fig 3: Gamma-ray transmissions (I/Io) for the same sample matrix (Ore-2) with different
densities (g/cm3) and for different photon energies.
22
1.0
Ore -1
Soil-1
Water
Ore-2
Energy, keV
59.5
81.2
276
302.4
356
383.9
661.6
1173.2
1332.5
0.9
I/Io
0.8
0.7
0.6
0.5
0.4
0.94
0.95
0.96
0.97
0.98
0.99
Denisty, g/cm
1.00
1.01
1.02
3
Fig 4: Gamma-ray transmissions (I/Io) for a set of different sample matrices with the
same densities, ±5% (g/cm3) and for different photon energies
23