J. Nig. Soc. Phys. Sci. 4 (2022) 621

Journal of the
Nigerian Society

of Physical
Sciences

Validation of Tritium Calibration Curve in CIEMAT/NIST
Activity Measurement Using Non Linear Least Squared Fittings

and Calculations of the Half-Life and Decay Constant of
Potassium-40

S. Adamsa,∗, E. Josepha, G. Kamalb

a Department of Physics, Federal University Dutsin-Ma, Katsina State-Nigeria.
b Department of Physics, Federal University, Lafia, Nasarawa State-Nigeria.

Abstract

Non-linear curve fitting of tracer efficiency is one of the uncertainty contributors in CIEMAT/NIST method for specific activity measurement and
half-life evaluation of radionuclide. This study applied least squared fitting of four different polynomials to validate the tracer efficiency calibration
curve for the specific activity measurements of nine Potassium Chloride samples from which half-life and decay constants were computed. All
samples were measured using TR1000 Liquid Scintillation Counter and the results of the relative standard uncertainties in tracer interpolated
efficiencies associated with the least square fit analysis were found to be 8.363%, 8.076%, 7.941% and 8.767% for polynomials of n = 2, n = 3,
n = 4 and n = 5, respectively. The corresponding values of 40K specific activity, from the application of empirical efficiencies generated with
these polynomials were found to be (16.541, 16.540, 16.537 and 16.548) Bq/g respectively. From these measured specific activity values, the
computed half-life and decay constants were found to be (1.2518, 1.2519, 1.2521 and 1.2513) ×109 y (for n = 2, n = 3, n = 4, and n = 5),
respectively, and (5.365, 5.5364, 5.5354 and 5.5391) ×10−10y−1 respectively. All values were found to be in good agreement with results of other
literatures. However to minimize uncertainty associated with empirically generated tracer interpolation efficiencies, least squared fit analysis of
tracer calibration curve should be done, with control trials using different polynomials so as to obtain the best fit.

DOI:10.46481/jnsps.2022.621

Keywords: CIEMAT/NIST method, Liquid Scintillation Counting, Least Square Fit, Analysis, Tracer Calibration Curve

Article History :
Received: 28 January 2022
Received in revised form: 3 May 2022
Accepted for publication: 5 May 2022
Published: 20 August 2022

c© 2022 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license

(https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Communicated by: W. A. Yahya

1. Introduction

Non-linear curve fitting is a very important component in
nuclear metrology [1]. In fact, it is applied in analytical formula
for half-life evaluation of radionuclide [2, 3] and one of the

∗Corresponding author tel. no: +2348132888358
Email address: silasadams50@gmail.com (S. Adams )

uncertainty contributors in CIEMAT/NIST method for specific
activity measurement [4]. This is because the CIEMAT/NIST
method relies on tracer non-linear curve fitting for accurate in-
terpolation of the relative counting efficiency of the radionu-
clide under investigation [5, 6]. The choice of both Tritium
and Potassium-40 (40K) in this study was because of the fol-
lowing reasons: (i) tritium suitability for use in CIEMAT/NIST

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S. Adams et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 621 2

method as a pure beta emitting radionuclide (ii) significance of
Potassium-40 (40K) in radiometric age determination [4] which
depends solidly on accurately determined half-life and decay
constants [7].
40K is a naturally occurring radionuclide aside uranium and tho-
rium [8], whose decay to Calcium-40 (one of the stable iso-
topes of Calcium) [9, 10] and especially to Argon-40 (figure 1)
is widely used in radiometric dating [4]. Unfortunately, un-
certainty in half-life and decay constant values have limited
present day radiometric dating [4]. More so, the recent at-
tribution of disagreement in the values of decay constants to
the method used in determining half-lives by Mc Donough et
al, [11] serves as a trigger for this study as slight uncertainty
of only one percent in decay constant could lead to signifi-
cant disagreements in the ages of radioisotopes [7]. Unlike,
geochronology, nuclear Physics experiments determine half-
life through activity measurement. CIEMAT/NIST efficiency
tracing method is one of such methods for specific activity mea-
surement whose worth has been proven in the activity mea-
surements of many radionuclide and employs fitting as the best
means of interpolation [4, 5, 6].
In partial response to the call made by half-life evaluators for
in-depth assessment of individual uncertainty components for
good half-life measurement [12], this paper therefore shows
how to validate the tracer (Tritium) calibration curve through
controlled trials of non-linear least square fit analysis with dif-
ferent polynomials to determine the relative uncertainty associ-
ated with interpolated counting efficiencies of tracer (Tritium).
This was achieved with the use of CIEMAT/NIST method. TR
1000 Liquid Scintillation Analyzer was used to measure the
specific activity of 40K in Potassium Chloride (KCl) samples
from which the half-life and decay constant were evaluated.

2. Materials and Methods

2.1. Sample Preparation and Counting

A total of nine (9) samples were prepared as follows: 15
ml of Ultima-Gold AB Scintillation cocktail was added each
to 9 glass vials(20 ml capacity) of low potassium content. In
a similar manner, 1 ml of Potassium chloride (KCl) solution
(5g in 25 ml of water) each was again added to all the glass
vials. All weighing with Mettler AC100 were gravimetrically
controlled. Samples were gradually quenched by adding vary-
ing but increasing amount of the nitromethane as quenching
agent then vigorously shaken and kept for stability. A blank
sample was also prepared without the addition of KCl salt. All
samples including blanks were carefully labelled and counted
using Tricarb TR 1000 LSA each for 60 minutes to obtain
good counting statistics. Records of count rates in counts per
minutes (CPM) and Spectral index of the samples (SIS) as
the quench indicating parameter (QIP) were taken. The total
counting time for potassium chloride samples was more than
30 days in order to obtain stable counts rates.

Figure 1: 4019 K simplied decay scheme [4]

2.2. Liquid Scintillation Techniques

Radiation emitted from dissolved radionuclide samples in
scintillation cocktails transferred energy to the organic scintil-
lator that in turn emits light photons. This way each emission
result is a pulse of light in form of digit called counts [13].

2.3. CIEMAT/NIST Model

The CIEMAT/NIST model used in this work for the calcu-
lation of 40K specific activity was modified from Broda et al.,
[14] as shown in figure 2. The slight modification (improve-
ment) is clearly reflected in figure 2 where four different poly-
nomials were used to generate the empirical efficiencies of the
tracer radionuclide (Tritium-3) using a method of interpolation
called least squared fit analysis as stated below on Non-linear
Least squared Fittings of Tritium calibration Curve. The usage
of these polynomials for the fittings is to account for the least
uncertainty associated with the empirically generated tracer ef-
ficiency as a result of the least squared fit analysis.

Broda et al description of CIEMAT/NIST method can be
summarized into five steps as follows: (i) theoretical computa-
tion of detection efficiencies of both tracer and the radionuclide
under study (40K in this case) . This is done via computer pro-
gram so as to establish an efficiency calibration curve (a plot
of nuclide efficiency as function of tracer efficiency) [14]. (ii)
Experimental determination of tracer efficiency from LSC mea-
surement (CPM and QIPs). Since the activity (DPM) of the
quench set is known, the efficiency of tracer is computed as
(CPM/DPM). (iii) Measurement of samples of radionuclide un-
der study for determination of tracer efficiencies and the corre-
sponding radionuclide efficiencies. By applying the QIPs ob-
tained from measured samples to equation (1) the tracer effi-
ciencies are obtained. Similarly, by applying the obtained tracer
efficiencies to equation (10), the corresponding radionuclide ef-
ficiencies are obtained. (iv) Conversion of the efficiency of the
radionuclide so determined and the measured count rate (Rn) in
CMP to the activity of the radionuclide [14].

Rn denotes sample count rate, µ is the free parameter, SIS
is the spectral index of the sample as quench indicating param-
eter, m, is the mass of KCl salt in-cooperated into the 1 ml

2



S. Adams et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 621 3

Figure 2: Illustration model for CIEMAT/NIST method [14].

Figure 3: Efficiency calibration curve (a plot of efficiency of 40K versus tracer efficiency)

of the sample εK−40and εH−3 efficiencies of Potassium-40 and
Tritium-3 radionuclides, respectively.

Normally, fittings of tracer calibration curves in
CIMAT/NIST method are done using a single order poly-
nomial [15]. However this study applied four (4) polynomials
of different order to validate the tracer calibration curve hence
generating four different sets of 40K efficiencies. This served
as the slight variation from other CIEMAT/NIST method.

2.4. Analysis

The significant analysis in this study is based on the non-
linear least squared fittings at different polynomial orders for
tracer (Tritium) calibration curve as well as the evaluation of
40K specific activity and half-life.

2.5. Non-linear Least squared Fittings of Tritium calibration
Curve

Four least squared fitted calibration curves for tracer effi-
ciency were obtained at varying polynomial order (n = 2, 3,
4, and 5) for tracer calibration curve (a plot of tracer efficiency
εH−3versus quench indicating parameter). The tracer calibra-
tion curve upon which the current least square is fitted was ob-
tained from radiation analysis [16]. The general form of poly-
nomial used to generate the empirical tracer efficiencies for the
least square fit analysis is given as equation (1) [17] while equa-
tions (2), (3), (4), and (10a) were used for the four different
non-linear least square fittings in accordance with the polyno-
mial order. All fittings were done using excel spread sheet in-
cluding the calculations of relative standard uncertainties in the
interpolated tracer efficiency associated with the least squared
fittings using equations (7), (8) and (9) as presented in table 1

3



S. Adams et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 621 4

(a) (b)

(c) (d)

Figure 4: Least squared fitted Tritium Calibration Curve for polynomials of order (a) n = 2; (b) n = 3; (c) n = 4 and (d); n = 5 respectively.

Figure 5: Combined least Squared Tritium calibration Curve for Polynomials
(n = 2, n = 3, n = 4, n = 5)

and figures 3, 4, 5, and 6.

εH−3 =

n∑
i=0

ki(S IS )
i (1)

For n = 2, the least squared theory becomes:

εH−3 = k0 + k1 (S IS ) + k2(S IS )
2 (2)

For n = 3 the least squared theory for fitting the tritium calibra-
tion curve becomes

εH−3 = k0 + k1 (S IS ) + k2(S IS )
2

+ k3(S IS )
3 (3)

Similarly for n = 4 and n = 5 the theory for the least squared
fitting of the calibration curve is given by

åH−3 = k0 +k1 (S IS )+k2(S IS )
2
+k3(S IS )

3
+k4(S IS )

4(4)
4



S. Adams et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 621 5

Table 1: Least Squared fit analysis, for Polynomials (n = 2, n = 3, n = 4, n = 5)

Experiment Predicted Least squared Predicted Predicted Predicted Least squared Predicted Least
squared

n = 2 n = 3 n = 4 n = 5
SIS Exp. ε(H-3)% ε(H-3)% Delta ε(H-3)% Delta squared ε(H-3)% Delta squared ε(H-3)% Delta

squared
18.6 68 68.39 0.15217 67.939 0.00361 67.962 0.0014 67.949 0.00259
16 64 62.936 1.13039 63.507 0.24285 63.197 0.64414 63.865 0.01806
14.8 58 58.607 0.36942 59.058 1.12063 59.028 1.05738 58.71 0.50466
13.6 52 53.134 1.28618 53.286 1.65585 53.457 2.12396 52.707 0.50069
12 48 44.055 15.55986 43.775 17.84978 43.928 16.57536 43.708 18.41697
11 38 37.347 0.42523 36.909 1.18919 36.913 1.18076 37.343 0.43059
10.5 29 33.696 22.05335 33.242 17.9988 33.16 17.31165 33.863 23.65655
9.2 23 23.271 0.07371 23.066 0.0044 22.878 0.01482 23.585 0.34245
8.5 18 17.101 0.80676 17.249 0.56265 17.183 0.6673 17.151 0.71978
8 13 12.456 0.2955 12.971 0.00082 13.124 0.01553 12.108 0.79495

Sum 42.15261 40.628616 39.59230 45.38733

Standard Uncertainty(Su) in ε (H-3)% 0.684370 0.671884 0.663260 0.710143
Order of polynomial Coefficients

k0 k1 k2 k3 k4 k5
n = 2 -88.89410 15.84820 -0.39742
n = 3 -61.11043 9.00344 0.14012 -0.013502
n = 4 -3.136618 -10.32740 2.48410825 -0.1360546 0.002333
n = 5 -104.8710 10.70044 2.43550 -0.387136 0.021350 -0.000419

Figure 6: Relative uncertainty in tritium interpolated efficiencies for different
polynomial

and

εH−3 = k0 + k1 (S IS ) + k2(S IS )
2 (5)

+ k3(S IS )
3

+ k4(S IS )
4

+ k5(S IS )
5,

where k0, k1, k2, k3, k4, and k5, are the coefficients of the least
squared fittings respectively, SIS= Spectral index of the sample
for tracer, and εH−3 = tritium counting efficiency

Delta = E x periment − T heory (6)

Sum = Sum o f Delta Squared (7)

Standard error = S QRT
(

sum
n ∗ (n − 1)

)
(8)

From the standard error, the relative standard uncertainties in
empirically generated tritium efficiencies was obtained using
equation (9) [18].

u(εH−3)
εH−3

=

√∑ ( S ε
εH−3

)2
(9)

2.6. Calculation of Efficiencies
All theoretical efficiencies for tracer (3H) and 40K were cal-

culated using CN2003 code. The range of the calculation cho-
sen for the tracer (Tritium) efficiency was between 25% to 65%
for ionization quench constant of 0.0075 cm/MeV. The interpo-
lated efficiency (εK−40)40K was obtained using a fitted polyno-
mial given by equation (10) from the general form of equation
given by [17] for some selected values of free parameter as
shown in figure 3. The CN2003 visual basic code takes into
account the following: statistics of nuclear decay, detector re-
sponse as well as Physics of the nuclear decay processes.

εK−40 =

n∑
i=0

ki(εH−3)
i (10a)

ε(K − 40)% = 2E − 05ε(H − 3)3 − 0.001ε(H − 3)2 (10b)
+ 0.081ε(H − 3) + 88.39

The regression value of 1 on figure 3 indicates a perfect
polynomial fitting.

2.7. 40K specific Activity and Half-life
The measured SIS for Potassium samples were applied to

the least squared fitted calibration curves to obtain the corre-
sponding tracer (Tritium) efficiencies. The corresponding val-
ues of 40K efficiencies (εK−40 ) were obtained by fitting the em-
pirically generated tracer efficiencies to equation (10b). Equa-
tion (10b) is the efficiency calibration curve extracted from
CIEMAT/NIST calculation for some selected values of free pa-
rameter (µ). With the efficiencies of 40K (εK−40 ) obtained using
equation (10b) and data of 40K measurement from Liquid Scin-
tillation counter (Count rates and SIS), the specific activities of
40K in all the measured samples were computed using equation
(11) as follows [4]:

a =
Rs − Rb
mεK−40

, (11)

5



S. Adams et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 621 6

Table 2: Specific activities, half-lives, decay constants and efficiencies for all KCl samples

Order of Polynomial Sample ID Efficiency Specific Activity Half-life
(
T1/2

)
Decay constant (λ)

�(H − 3)% �(K − 40)% Bq/g ×109 y ×10−10y−1

n = 2 GVT1 60.6755 94.0907 12.916 1.6031 4.3234
GVT2 49.764 92.4092 16.247 1.2745 5.4344
GVT3 46.6121 92.0183 17.011 1.2173 5.6908
GVT4 46.7852 92.0388 17.651 1.1731 5.9048
GVT5 50.6193 92.5219 15.839 1.3073 5.2981
GVT6 46.2459 91.9753 18.741 1.1049 6.2697
GVT7 49.2336 92.3407 16.556 1.2507 5.5381
GVT8 47.2002 92.0884 16.637 1.2447 5.5555
GVT9 48.5306 92.2517 17.376 1.1917 5.8125

Mean 16.541 1.2518 5.5365
n = 3 GVT1 61.2099 94.1879 12.91 1.6047 4.3189

GVT2 49.7335 92.4052 16.236 1.2754 5.4346
GVT3 46.432 91.9971 17.005 1.2177 5.692
GVT4 46.6126 92.0184 17.645 1.1736 5.9061
GVT5 50.6334 92.5237 15.828 1.3083 5.298
GVT6 46.0501 91.9525 18.736 1.1052 6.271
GVT7 49.1763 92.3334 16.546 1.2514 5.5385
GVT8 47.046 92.0699 16.63 1.2452 5.5562
GVT9 48.4386 92.2402 17.367 1.1923 5.8132

Mean 16.540 1.2519 5.5364
n = 4 GVT1 61.0546 94.1595 12.907 1.6043 4.3204

GVT2 49.2672 92.345 16.232 1.2757 5.4331
GVT3 46.168 91.9662 17.002 1.2179 5.6908
GVT4 46.336 91.9858 17.651 1.1738 5.9047
GVT5 50.1258 92.4565 15.834 1.3086 5.2965
GVT6 45.8133 91.9251 18.731 1.1055 6.2699
GVT7 48.7391 92.2779 16.542 1.2518 5.537
GVT8 46.7397 92.1912 16.545 1.2478 5.5547
GVT9 48.04352 92.1912 17.363 1.1926 5.8117

Mean 16.537 1.2521 5.5354
n = 5 GVT1 61.1382 94.1748 12.905 1.6045 4.3195

GVT2 49.2672 92.34508 16.247 1.2745 5.4388
GVT3 46.168 91.96626 17.011 1.2173 5.6941
GVT4 46.336 91.98589 17.651 1.1731 5.9082
GVT5 50.1258 92.45652 15.839 1.3073 5.3019
GVT6 45.8133 91.92513 18.741 1.1049 6.2731
GVT7 48.7391 92.27796 16.556 1.2507 5.5418
GVT8 46.7397 92.03347 16.637 1.2447 5.5592
GVT9 48.0435 92.19121 17.376 1.1917 5.5816

Mean 16.548 1.2513 5.5391

where Rs =Sample count rate, Rb = Background count rate,
m = mass of potassium salt incorporated into the sample,
εK−40 = counting efficiency for 40 K.

From the specific activity obtained using equation (8) the
half-life of 40 K was computed using equation (12) [4]:

T1/2 =
(ln 2) NAM p

a
, (12)

where NA = Avogadro constant, M is the relative molar mass of
KCl, p = N(4019 K)/N (K) = 0.01167%.

Following the relationship between half-life and decay con-

stant, the decay constant was computed using equation (13) ex-
pressed as [11]:

λ =
ln 2
T1/2

(13)

3. Results and Discussion

Results of Least Squared fitted Tritium Calibration Curves
for all the polynomials is presented in figures 4 and 5.

From figure 4 (a, b, c and d), the least squared fitted curves
of the experimental tritium calibration curve it can be observed

6



S. Adams et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 621 7

Table 3: 40KHalf-life and decay constant values from source and some calcu-
lated decay constants

Reference 40K (T1/2) ×10
9y 40K (λ) ×10−10y−1

Original Original Calculated

Steiger and Jägar,
(1977) [20]

1.250 5.543 -

Min et al. (2000)
[19]

1.269 5.463 -

Grau Malonda
and Grau Carles
(2002) [21]

1.248 - 5.554

Kossert and
Günther,(2004)
[4]

1.248 - 5.554

This study (polynomial order)
n = 2 1.2519 5.5367 5.5367
n = 3 1.2518 5.5365 5.5365
n = 4 1.2521 5.5354 5.5354
n = 5 1.2513 5.5391 5.5391

that not all points of empirically generated efficiencies are per-
fectly fitted to the experimental data. This is because each fitted
polynomial has an associated level of uncertainty in determin-
ing the empirical efficiencies for interpolation with the spec-
tral index of Potassium Chloride (KCL) measured samples as
clearly shown in figure 5.

The magnitude of the standard uncertainties associated with
each polynomial including the relative standard interpolated ef-
ficiencies for the least squared analysis is shown is Table 1.

All least squared fit analysis in this study and the estima-
tion of standard and relative interpolated efficiencies of Tri-
tium were carried out using Excel spread sheet. From table
1 it can be seen that the standard uncertainties and relative
standard uncertainties in tritium interpolated efficiencies due to
the Least squared fitted polynomials ranges from (0.663260 to
0.710143)% and (7.941 to 8.767)% respectively. Polynomial of
order (n = 4) has the lowest interpolation uncertainties while
the highest uncertainty was associated to polynomial of n=5 as
clearly shown in figure 6. It can also be deduced from figure
5 that the uncertainties associated with non-linear least square
fit analysis of the interpolated efficiencies does not completely
depend on the polynomial order.

3.1. 40K Specific Activity, Half-life and Decay Constant

The values of 40K specific activity, as well as the computed
half-life and decay constants obtained in this study for the four
(4) different polynomials are presented in table 2.

From results presented in table 2, it can be seen that the
mean obtained values of 40K specific activity, Half-life and de-
cay constant for the four least squared fitted polynomials var-
ied slightly ranging from (16.537 to 16.548) Bq/g, (1.2513
to1.2521) ×109 y and (5.5354 to 5.5392)×10−10y−1 respec-
tively. This slight variation can be linked to the relative standard
uncertainties associated with empirically generated interpola-
tion efficiencies of the tracer (Tritium) from the least squared

fitted polynomials. Polynomial of n=4 has the lowest spe-
cific activity value of 16.537 Bq/g followed by Polynomial of
n = 3 with 16.540 Bq/g, then polynomial of n = 2 (which
is basically quadratic) with 16.541 Bq/g and lastly polynomial
of n = 5 with the highest value of16.548.This results shows
that the higher the relative uncertainty associated with the least
squared polynomial fitting, the higher the values of specific ac-
tivities and decay constants. On the contrary, higher values of
half-life are linked to lower relative standard interpolated un-
certainties of tracer (Tritium) associated with the least squared
fitted polynomials. However, all values of specific activities in
this current study are in good agreement with values of (16,594
to 16.616) Bq/g obtained by Kossert and Günther, [4] while the
half-life and decay constant values are in good agreement with
the values obtained by Steiger and Jäger, Min et al., Grau Mal-
onda and Grau Carles, Kossert and Güther, [4, 19, 20, 21] as
shown in table 3.

4. Conclusion

The relative standard uncertainties associated with the in-
terpolated tracer efficiencies from the least squared fitting of
the four different polynomials used in this study revealed that
the choice of the prefer polynomial fit does not necessary de-
pend on the order of polynomial but on the coefficients of the
polynomial that will produce the lowest standard error. This is
obtained through control trials starting from the lowest order.
The validity of this is seen in the slight variation of 40K specific
activity, upon which half-life is computed as well as decay con-
stant. The slight variation in these values is as a result of the
closed agreement in the relative standard uncertainties of the
fitted polynomials. However a more noticeable variation is ob-
served with least squared fitted polynomial of n=5. The choice
of the best fit will go a long way in further reducing the rel-
ative uncertainty associated with interpolated tracer efficiency,
which could improve the accuracy of CIEMAT/NIST method
in the activity determination of radionuclide and strengthen its
acceptance. Also, further non-linear least square fit analysis
of tracer (Tritium) calibration curve should be carried out with
polynomials of higher order to confirm the independence of the
best fit on the order of polynomial.

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