Adv. Studies Theor. Phys., Vol. 4, 2010, no. 14, 665 - 678
The Effect of the Activation Energy, Frequency
Factor and the Initial Concentration of Filled
Traps on the TL Glow Curves of
Thermoluminescence
Md. Shah Alam1,2 and Sabar Bauk1
1
Physics Section, School of Distance Education
Universiti Sains Malaysia, 11800 Penang, Malaysia
[email protected] 2
Department of Physics
Shahjalal University of Science and Technology
Sylhet, Bangladesh
Abstract
We have represented different types of models for the explanation of
thermoluminescence glow curves such as Randall-Wilkins model, Garlick-Gibson
model and May-Partridge model. We have studied the effect of the activation
energy, frequency factor and the initial concentration of filled traps on the glow
curves using MATHEMATICA for the above mentioned models.
PACS: 78.60.Kn
Keywords: Thermoluminescence glow curves, activation energy, frequency
factor, concentration of filled traps
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Md. Shah Alam and S. Bauk
1. Introduction
Thermoluminescence (TL) is the thermally stimulated emission of light following
the previous absorption of energy from radiation. The typical glow curve contains
one or more glow peaks. Each peak gives information about each trap level and its
occupation state, etc in the TL material. The glow peak is analyzed by an empirical method in which a parameter called the order of kinetics is introduced. When
the trapped electrons jump up to the conduction band by the thermal energy, they
have two kinds of chances to jump down. One is the retrapping process returning
to the same kind of traps and another is the recombination with the hole
accompanied by the emission of TL light. When the probability of being retrapped
is negligible, the glow curve has a narrow peak shape by a rapid recombination
process explained by Randall and Wilkins in 1945 [1]. Instead, if the retrapping
dominates, the recombination with the holes is suppressed and the curve has a
wide peak explained by Garlick and Gibson in 1948 [2]. These two descriptions
are called the first order kinetics and the second order kinetics respectively.
Between these two types, the general order kinetics is introduced for providing a
proper analytic continuation from the discrete two types of kinetics explained by
May and Partridge in 1964 [3]. In the second section we will clearly explain these
three models i.e., Randall-Wilkins model, Garlick-Gibson model and MayPartridge model. We explain the effect of the activation energy on the glow curves
in section three, the effect of the frequency factor on the glow curves in section
four and the effect of the initial concentration of filled traps on the glow curves in
section five using MATHEMATICA for the above mentioned models.
2. Different Models of Thermoluminescence
2.1 Randall and Wilkins Model
CB
E
Electron Trap, Tr
R
VB
Figure 1: Randall Wilkins Model
We have found the Randall and Wilkins model for the thermoluminescence in
different books [4-6] and journals [7-9]. In 1945, Randall and Wilkins extensively
used a mathematical representation for each peak in a glow curve, starting from
studies on phosphorescence. Their mathematical treatment was based on the
energy band model and yields the well-known first order expression. Figure 1
Thermoluminescence glow curves
667
shows the simple model used for the theoretical treatment. Between the delocalized bands, conduction band CB, and valence band VB, two localized levels
(metastable states) are considered, one acting as a trap Tr, and the other acting as
a recombination center R. The distance between the trap Tr and the bottom of the
CB is called the activation energy or trap depth E. This energy is the energy
required to liberate a charge, i.e. an electron, which is trapped in Tr. According to
this model, the TL intensity I, at a constant temperature can be written as
………………………………….. (1)
where n (cm ) is the electron concentration trapped at time t(s), k (eVK−1) is the
Boltzmann’s constant, S(sec−1) is the frequency factor and T(K) is the
temperature.
If the sample is heated up so that the temperature rises at a linear heating rate
−3
−1
(K s ),
then
exp
………………………………….. (2)
On integration of Eq. (2), one obtains
exp
/
/
………………………………….. (3)
where n0 (cm ) is the concentration of traps populated at the starting heating
temperature T0 (K). The temperature dependence of the emitted TL is given by
−3
/
/
………………………………….. (4)
2.2 Garlick-Gibson Model
Garlick and Gibson [2] considered the case of strong retrapping, which brought
about the second-order kinetics. According to this model, the TL intensity, I, at a
constant temperature can be written as
………………………………….. (5)
−3
where N (cm ) is the trap concentration. The equation describing the TL
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Md. Shah Alam and S. Bauk
intensity for second-order glow peak is given by [2]
1
exp
………………………………….. (6)
2.3 May-Partridge Model
May and Partridge [3, 11] and others have proposed an empirical equation to
describe the TL glow peak when conditions for neither first-order nor secondorder are satisfied. This equation is known as the general-order kinetics and its
final form has been suggested as [12]
exp
/
………………………………….. (7)
where b is the order of kinetics. Usually b is assumed to be between 1 and 2, but
sometimes can exceed this range [13]. The solution of Eq. (7) for b ≠ 1 is given by
[12]
"
1
1 "/
exp
/
………………………………….. (8)
where
"
/
.
………………………………….. (9)
3. The Effect of The Activation Energy on The TL Glow Curves
3.1 The effect of the activation energy on the TL glow curves for the Randall
and Wilkins model
Using MATHEMATICA we can plot the TL glow curves for different activation
energies. Here is an example of three Randall and Wilkins model (first order) TL
curves, calculated using three different values of the activation energies E1=1.0
eV, E2=1.5 eV and E3=2.0 eV. All other parameters are the same i.e., frequency
factor, s =1012 sec−1 and Boltzmann’s constant, k= 8.617×10−5 eVK−1 and the
−3
initial concentration of filled traps, n0 = 1010 cm . We get the curves as shown in
Figure 2.
Thermoluminescence glow curves
669
E1 = 1.0 eV
TL T TL glowcurves
: E1=1.0, E2=1.5, E3=2.0 E2
8
3ґ10
= 1.5 eV
E3 = 2.0 eV
8
2.5 ґ10
2ґ108
8
1.5 ґ10
1ґ108
7
5ґ10
T 100 200
300
400
500
600 Figure 2: Randall and Wilkins model TL glow curves for the activation energies
1.0 eV, 1.5eV and 2.0 eV.
3. 2 The effect of the activation energy on the TL glow curves for the GarlickGibson model
Here is an example of three Garlick-Gibson model (second order) TL curves,
calculated using three different values of the activation energies E1=1.0 eV,
E2=1.5 eV and E3=2.0 eV. All other parameters are the same i.e., frequency
factor, s =1012 sec−1 and Boltzmann’s constant, k= 8.617×10−5 eVK−1 and the
−3
initial concentration of filled traps, n0 = 1010 cm . We get the curves as shown in
Figure 3.
E1 = 1.0 eV
TL T TL glow curves
: E1=1.0, E2=1.5, E3=2.0
E2 = 1.5 eV
2 ґ10 8 E3 = 2.0 eV
8 1.5 ґ10 1 ґ10 8 5 ґ10 7 T 100 200 300
400
500
600
Figure 3: Garlick-Gibson model TL glow curves for the activation energies 1.0
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Md. Shah Alam and S. Bauk
eV, 1.5 eV and 2.0 eV.
3.3 The effect of the activation energy on the TL glow curves for the MayPartridge model
Here is an example of three May-Partridge model (general order) TL curves,
calculated using three different values of the activation energies E1=1.0 eV,
E2=1.5 eV and E3=2.0 eV. All other parameters are the same i.e., frequency
factor, s =1012 sec−1 and Boltzmann’s constant, k= 8.617×10−5 eVK−1 and the
−3
initial concentration of filled traps, n0 = 1010 cm . We get the curves as shown in
Figure 4.
E1 = 1.0 eV
TL T
TL glow curves
: E1=1.0, E2=1.5, E3=2.0
E2 = 1.05eV
8 1 ґ10 E3 = 2.0 eV
7 8 ґ10 7 6 ґ10 7 4 ґ10 2 ґ10 7 200
400 600
800
1000
T 1200 Figure 4: May-Partridge model TL glow curves for the activation energies 1.0 eV,
1.5 eV and 2.0 eV.
Remarks: As the activation energy E is increased, the TL glow curve shifts
towards higher temperatures, but the curve maintains its overall shape for these
three models.
Thermoluminescence glow curves
671
4. The Effect of The Frequency Factor on The TL Glow Curves
4.1 The effect of the frequency factor on the TL glow curves for the Randall
and Wilkins model
Here is an example of three Randall and Wilkins model (first order) TL curves,
calculated using three different values of the frequency factors, s1 = 108 sec−1, s2
=1010 sec−1 and s3 =1012 sec−1. All other parameters are the same i.e., activation
energy, E = 1.0 eV and Boltzmann’s constant, k = 8.617×10−5 eVK−1 and the
−3
initial concentration of filled traps, n0 = 1010 cm . We obtained the curves in
Figure 5.
s3 = 1012 sec−1
TL T TL glow curves : s1=10^8, s2=1010
^10, s3=10^12
s2 = 10 sec−1
8 3 ґ10 s1 = 108 sec−1
8 2.5 ґ10 2 ґ10 8 8 1.5 ґ10 1 ґ10 8 7 5 ґ10 T 100 200
300
400
Figure 5: Randall and Wilkins model TL glow curves for the frequency factors, s1
= 108 sec−1, s2 =1010 sec−1 and s3 =1012 sec−1.
4.2 The effect of the frequency factor on the TL glow curves for the GarlickGibson model
Here is an example of three Garlick-Gibson model (second order) TL curves,
calculated using three different values of the frequency factors, s1 = 108 sec−1, s2
=1010 sec−1 and s3 =1012 sec−1. All other parameters are the same i.e., activation
energy, E =1.0 eV and Boltzmann’s constant, k= 8.617×10−5 eVK−1 and the initial
−3
concentration of filled traps, n0 = 1010 cm . We get the curves as shown in Figure
6.
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Md. Shah Alam and S. Bauk
s3 = 1012 sec−1
TL T TL glow curves : s1=10^8,
10sec
^10
−1, s3=10^12
s2 =s2
10=10
2 ґ10 8 8
s1 = 10 sec−1
8 1.5 ґ10 1 ґ10 8 5 ґ10 7 T 100 200
300
400
Figure 6: Garlick-Gibson model TL glow curves for the frequency factors, s1 =
108 sec−1, s2 =1010 sec−1 and s3 =1012 sec−1.
4.3 The effect of the frequency factor on the TL glow curves for the MayPartridge model
Here is an example of three May-Partridge model (general order) TL curves,
calculated using three different values of the frequency factors, s1 = 108 sec−1, s2
=1010 sec−1 and s3 =1012 sec−1. All other parameters are same i.e., activation
energy, E = 1.0 eV and Boltzmann’s constant, k = 8.617×10−5 eVK−1 and the
−3
initial concentration of filled traps, n0 = 1010 cm . We get the curves as shown in
Figure 7.
Thermoluminescence glow curves
673
s3 = 1012 sec−1
TL T
TL glowcurves : s1=10^8, s2=10^10, s3=10^12
s2 = 1010 sec−1
8 1 ґ10 7 8 ґ10 s1 = 108 sec−1
7 6 ґ10 7 4 ґ10 2 ґ10 7 200
400 600
800
1000
T 1200 Figure 7: May-Partridge model TL glow curves for the frequency factors, s1 = 108
sec−1, s2 =1010 sec−1 and s3 =1012 sec−1.
Remarks: As the frequency factor s is increased, the TL glow curve shifts
towards lower temperatures, but the curve maintains its overall shape for these
three models.
5. The Effect of the Initial Concentration of Filled Traps on the
TL Glow Curves
5.1 The effect of the initial concentration of filled traps on the TL glow curves
for the Randall and Wilkins model
Here is an example of three Randall and Wilkins model (first order) TL curves,
calculated using three different values of the initial concentration of filled traps, n0
−3
= 1 × N, 0.5 × N and 0.1 × N (where N = 1010 cm ). All other parameters are the
same i.e., activation energy, E = 1.0 eV and Boltzmann’s constant, k = 8.617×10−5
eVK−1 and the frequency factor, s = 1012 sec−1. We get the curves as shown in
Figure 8.
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Md. Shah Alam and S. Bauk
n0 = 1×1010
TL T 1
First order kinetics
: no=1* N, 0.5* N, 0.1* N
8 3 ґ10 n0 = 0.5×1010
8 2.5 ґ10 1
2 ґ10 8 8 1.5 ґ10 n0 = 0.1×1010
1
1 ґ10 8 7 5 ґ10 T 50 100
150
200
Figure 8: Randall and Wilkins model TL glow curves for initial concentration of
−3
−3
−3
filled traps, n0 = 1×1010 cm , 0.5×1010 cm and 0.1×1010 cm .
Remarks: The initial concentration of filled traps (no) affects only the maximum
height of the TL glow curve, and leaves the shape of the 1st order TL glow curve
and the temperature of maximum TL intensity (Tmax) unchanged.
5.2 The effect of the initial concentration of filled traps on the TL glow curves
for the Garlick-Gibson model
Here is an example of three Garlick-Gibson model (second order) TL curves,
calculated using three different values of the initial concentration of filled traps, n0
−3
= 1 × N, 0.5 × N and 0.1 × N (where N = 1010 cm ). All other parameters are the
same i.e. activation energy, E = 1.0 eV and Boltzmann’s constant, k = 8.617×10−5
eVK−1 and the frequency factor, s = 1012 sec−1. We get the curves as shown in
Figure 9.
Thermoluminescence glow curves
675
n0 = 1×1010
1
TL T Second order kinetics
: no=1* N, 0.5* N,
0.1* N
2 ґ10 8 n0 = 0.5×1010
1
8 1.5 ґ10 n0 = 0.1×1010
1 ґ10 8 1
5 ґ10 7 T 50 100
150
200
Figure 9: Garlick-Gibson model TL glow curves for initial concentration of filled
−3
−3
−3
traps, n0 = 1×1010 cm , 0.5×1010 cm and 0.1×1010 cm .
Remarks: The initial concentration of filled traps (no) affects both the maximum
height of the TL glow curve and the temperature of maximum TL intensity (Tmax),
but leaves the overall shape unchanged.
5.3 The effect of the initial concentration of filled traps on the TL glow curves
for the May-Partridge model
Here is an example of three May-Partridge model (general order) TL curves,
calculated using three different values of the initial concentration of filled traps, n0
−3
= 1 × N, 0.5 × N and 0.1 × N (where N = 1010 cm ). All other parameters are the
same i.e. activation energy, E = 1.0 eV and Boltzmann’s constant, k = 8.617×10−5
eVK−1 and the frequency factor, s = 1012 sec−1. We get the curves as shown in
Figure 10.
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Md. Shah Alam and S. Bauk
n0 = 1×1010
TL T
Generalorder kinetics
: no=1* N, 0.5* N, 10.1* N
8 1 ґ10 7 8 ґ10 n0 = 0.5×1010
1
7 6 ґ10 n0 = 0.1×1010
1
7 4 ґ10 2 ґ10 7 T 100 200
300
400
500
Figure 10: May-Partridge model TL glow curves for initial concentration of filled
−3
−3
−3
traps, n0 = 1×1010 cm , 0.5×1010 cm and 0.1×1010 cm .
Remarks: The initial concentration of filled traps (no) affects both the maximum
height of the TL glow curve and the temperature of maximum TL intensity (Tmax),
but leaves the overall shape unchanged.
6. Conclusion
We have studied the effect of the activation energy, frequency factor and the
initial concentration of filled traps on the glow curves using Randall-Wilkins
model, Garlick-Gibson model and May-Partridge model. We have observed the
following results:
(i) As the activation energy E is increased, the TL glow curve shifts towards
higher temperatures, but the curve maintains its overall shape for these three
models.
(ii) As the frequency factor s is increased, the TL glow curve shifts towards lower
temperatures, but the curve maintains its overall shape for these three models.
(iii) The initial concentration of filled traps (no) affects only the maximum height
of the TL glow curve, and leaves the shape TL glow curve and the temperature of
maximum TL intensity (Tmax) unchanged in the case of the Randall-Wilkins
model.
(iv) The initial concentration of filled traps (no) affects both the maximum height
of the TL glow curve and the temperature of maximum TL intensity (Tmax), but
Thermoluminescence glow curves
677
leaves the overall shape unchanged in the case of the Garlick-Gibson model and
May-Partridge model.
References
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traps. I. The study of trap distributions. Proc. Roy. Soc. Lond. A 184, 366.
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luminescence in sulphide and silicate phosphors. Proc. Phys. Soc. 60, 574.
[3] May, C. E. and Partridge, J. A. (1964). Thermoluminescence kinetics of alpha
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[11] Partridge, J. A. and May, C. E. (1965). Anomalous thermoluminescence
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[13] Chen, R. (1969). Glow curves with general order kinetics. J. Electrochem.
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Received: May, 2010