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Computer analysis of the thermoluminescence glow peaks
Journal of Luminescence 11(1976) 357—361 © North-holland Publishing Company
COMPUTER ANALYSIS OF THE THERMOLUMINESCENCE GLOW PEAKS A. SATHYAMOORTHY, K.C. BHALLA and J.M. LUTHRA Chemistry Division, Bhabha Atomic Research Centre, Trombay, Bombay 400 085, India Received 13 September 1975
A computational procedure for the analysis of thermoluminescence (TL) glow peaks based on a simple model for TL, assuming transitions via the conduction band has been described. The method takes into consideration the full shape of the glow peak and utilizes the curve fitting technique and a general order kinetic equation. A consistent value for trap depth (E) corresponding to the observed peak temperature (Tm) is first calculated forchosen values of other parameters and is used to generate a theoretical glow curve which is then compared with the experimental one. By consideration of the various restraints the parameters are altered to obtain a good fit. A set of parameters which produce a glow curve differing only slightly from the experimental one in the least-square sense is taken as the final set to describe the TL process. TL parameters obtained in the case of MgO by application of this method are reported.
1. Introduction Thermoluminescence (TL) has become a convenient tool for studying traps in crystal. Generally it involves radiative recombination of thermally released charge carriers from their traps with complementary charges at the recombination centres. Analysis of TL peaks provides information on recombination kinetics, which is essential for complete understanding of the TL process. Many methods have been advocated in the literature by several authors for the purpose. These methods have been reviewed by Braunlich [11and Chen [2]. Most of these methods rely rather heavily on the peak temperature and some other special features of the glow peak (such as half-width, full-width, lower and higher half-width temperatures) and have limited applicability. No general method is as yet available which is applicable to all cases of TL. As the observed TL is the net result of multiple processes taking place on heating the material, methods based on the shape of the entire glow peak are more useful than others. The method advocated by Maxia et al. [3] can be classified in this category. It is a generalised initial rise method of analysis by which trap parameters can easily be obtained for saturated glow peaks. This paper describes a computational procedure using the entire glow peak and is applicable to a single glow peak. 357
358 A. Sathyamoorthy ci al.
/ Computer analysis of the thermoluminescence glow
peaks
2. Model The model (fig. I) which is used by many workers in this field, assumes that recombination transitions responsible for TL emission take place via the conduction band. No direct transitions between the traps and recombination centres or between the defects and valence bands are considered. The basic equations governing such a recombination process are given by Braunlich [1] as:
dn/dtcrh—f3n(H--h)—ynf, dh/dt= --cth+13n(H-- h),
f= n + h, 1(T) =-~nf, where n = the density of electrons in conduction band, h = the density of trapped electrons at temperature T, f the density of unoccupied recombination centres, H = the density of traps available, 1(T) = the TL intensity at temperature T, cs = the coefficient for thermal release of trapped charges, ~3= the coefficient for retrapping transition, y = the coefficient for recombination. These equations can be solved using an approximation that n is small compared to h and further that it remains constant during the TL process i.e. dn/dt 0 andf~h. This leads to a general solution for TL intensity: —
d.h dt
2et/T Ha R+(1 0b .-R)b’
with 1
b
1
R R
Ib\ \b 0/
1
~
b0
~
T
forR*0,
GB
/~
n ~ H
j~
A
Fig. 1. Simple model of TL process.
~E
(2a)
A. Sathyamoorthy et al.
/ c’omputer analysis of the thermoluminescence glow peaks 359
and
~ where R
~fe~t/TdT
=
~/y
=
forRO,
(2b)
the retrapping factor, a
0 = the frequency factor, b = h/H = occupancy, the constant heating rate, k = Boltzmann constant, F = trap depth, ~ = E/k. T0 = the temperature at which TL glow starts, b0 = the value of b at T = T0. The cases where R = 0 and R = 1 give the well-known first and second order kinetic equations of Randall and Wilkins [4] and Garlick and Gibson [5] respectively. These simple equations are good approximations to start with for glow curve analysis even for cases where more complex kinetics may govern the process. In fact, many subsequent theoretical improvements in TL theory were based on these simple cases in one form or another. Two important points about this model are to be noted: (i) The peak intensity is predicted at temperatures T10 where the followirit, relation holds between the various quantities. q
=
a0 bme_t/Tm ~~T~R+(l~
Reff,
(3)
where Reii
[ —
[2
—
(l—R)b01 R+(l R)b
may be called the effective kinetic order (note: limR~lReff = 2 and limR..+oReff The subscript m denotes the values of the quantities at T = Tm. (ii) Takingq dt = dT, the integration of first two terms of eq. (1) gives
h—h0
=
—(1/q)S(T),
where
S(fl=fI(fldT which can be rewritten as
b—b0=KS(T),
(4)
where the constant K depends on heating rate, H values and the light collection efficiency of the TL set up. This shows that the exact b’s versus fractional areas S(T)’s gives straight line.
360 A. Sathyamoorthy et al. / Computer analysis of the thermoluminescence glow peaks 3. Computational procedure Since the observed TL intensities are generally in arbitrary units, it is desirable to compare only the relative intensities I(T)/I(Tm) for both the observed and theoretical glow curves. In principle a computed curve which fits with the experimental one in the ‘least-square sense” can be selected by suitably modifying the parameters, such that the quantity
6
=
2
~ [J~XP(T) JC~P(T) 1~’~(T~) JCal(T)J 1
is a minimum. Instead of arbitrarily adjusting the parameters by iteration so as to satisfy this condition, we have considered the constraints imposed on the values of the parameters by eqs. (3) and (4). The procedure adopted for this purpose is described below. Taking the experimental value of q, T~’1= T~P= Tm and the approximate values of a 0, R and b0 non-linear eqs. (2) and (3) are solved simultaneously by iterative methods to get the consistent ~ and bm values. Using (2) the various b’s are then computed for the temperature range considered and the correlation coefficient p between these b’s and the corresponding fractional areas S(fl’s of the experimental curve is calculated (for a set of n values ofy f(x), p being defined as: 2 (x~— x5)(y1 — Ya)1 — — xa)2 ~(Y. — Ya)2
[~I~
whereya = ~iyiIn andxa = For a straight line the absolute value of p is I). The b
0 is then altered so as to increase p and the process is repeated till the change (~b0)in b0 between two consecutive cycles ~0.00l. The value of p was found to be of the order of 0.98 thereby showing a very good linear relationship between b’s and S(T)’s. Of the remaining parameters a0 and R, a0 was fixed and R changed to locate minimum in 6 by repeating the entire above mentioned process for eachR, till LXR ~0.00l. Finally, a0 was altered by a factor of 10 and the entire scheme repeated. The final set of a0, R, with the corresponding ~ value giving the least value of 6 was taken as the best set fitting the experimental glow curve. 4. Application This method of analysis has been applied to determine 60Co) the TLFe3~(0.0lMole%) parameters for 380 K glow peak observed in y-irradiated (dose 0.1 M rad doped MgO. The peak is well isolated except for a slight overlapping from the higher temperature peak. This peak was completely isolated from other peaks by the cleaning
A. Sathyamoorthy et a!.
/ Computer analysis of the ther,noluminesc~nceglow peaks 361
Table 1 Influence of initial trap occupancy (b
10
0) on TL peak Tm
0.1 0.3 0.5 0.7
380.0 K, R
1.0 o~j= 10
bm
~ K
6/n
p
0.0544 0.1622 0.2699 0.3773
8407.70 8805.05 8990.64 9112.90
0.02768976 0.02329216 0.02125374 0.02006609
0.986305 0.987829 0.988453 0.988855
Is.
technique recommended by Hoogenstraaten [6]. A computer fit was obtained using various sets of TL parameters. Table 1 shows the influence of initial trap occupany (b0) for one such set as measured by 6 and p. The consistent b1~and ~ values are also tabulated for each set. It is seen that either 6 and p can be used as a criterion for choosing the best fit. All the computational work was carried out on a BESM-6 computer. The parameter set giving the best fit (with p = 0.9996 13 and 6 = 0.040344) was: 10/s, R = 0.1004, E0.796eV. b0 = 1.0, a0 = l0 This method has also been applied to determine the TL parameters in case of natural calcite. That study and computer programme will be published as BARC report.
Acknowledgement Authors thank Dr. M.D. Karkhanavala for useful suggestions.
References [11 P. Braunlich, Thermoluminescence of geological materials, ed. D.J. McDougall (Academic Press, New York, 1968) p.61. [2J R. Chen, J. App!. Phys. 40 (1969) 570. [3] V. Maxia, S. Onnis and A. Rucci, J. Luminescence (1971) 378. [41 J.T. Randall and W.H.F. Wilkins, Proc. Roy. Soc. A 184 (1945) 366. [5] G.I.J. Garlick and A.F. Gibson, Proc. Phys. Soc. A 60 (1948) 574. [6] W. Hoogenstraaten, Philips Res. Rep. 13 (1958) 515.
COMPUTER ANALYSIS OF THE THERMOLUMINESCENCE GLOW PEAKS A. SATHYAMOORTHY, K.C. BHALLA and J.M. LUTHRA Chemistry Division, Bhabha Atomic Research Centre, Trombay, Bombay 400 085, India Received 13 September 1975
A computational procedure for the analysis of thermoluminescence (TL) glow peaks based on a simple model for TL, assuming transitions via the conduction band has been described. The method takes into consideration the full shape of the glow peak and utilizes the curve fitting technique and a general order kinetic equation. A consistent value for trap depth (E) corresponding to the observed peak temperature (Tm) is first calculated forchosen values of other parameters and is used to generate a theoretical glow curve which is then compared with the experimental one. By consideration of the various restraints the parameters are altered to obtain a good fit. A set of parameters which produce a glow curve differing only slightly from the experimental one in the least-square sense is taken as the final set to describe the TL process. TL parameters obtained in the case of MgO by application of this method are reported.
1. Introduction Thermoluminescence (TL) has become a convenient tool for studying traps in crystal. Generally it involves radiative recombination of thermally released charge carriers from their traps with complementary charges at the recombination centres. Analysis of TL peaks provides information on recombination kinetics, which is essential for complete understanding of the TL process. Many methods have been advocated in the literature by several authors for the purpose. These methods have been reviewed by Braunlich [11and Chen [2]. Most of these methods rely rather heavily on the peak temperature and some other special features of the glow peak (such as half-width, full-width, lower and higher half-width temperatures) and have limited applicability. No general method is as yet available which is applicable to all cases of TL. As the observed TL is the net result of multiple processes taking place on heating the material, methods based on the shape of the entire glow peak are more useful than others. The method advocated by Maxia et al. [3] can be classified in this category. It is a generalised initial rise method of analysis by which trap parameters can easily be obtained for saturated glow peaks. This paper describes a computational procedure using the entire glow peak and is applicable to a single glow peak. 357
358 A. Sathyamoorthy ci al.
/ Computer analysis of the thermoluminescence glow
peaks
2. Model The model (fig. I) which is used by many workers in this field, assumes that recombination transitions responsible for TL emission take place via the conduction band. No direct transitions between the traps and recombination centres or between the defects and valence bands are considered. The basic equations governing such a recombination process are given by Braunlich [1] as:
dn/dtcrh—f3n(H--h)—ynf, dh/dt= --cth+13n(H-- h),
f= n + h, 1(T) =-~nf, where n = the density of electrons in conduction band, h = the density of trapped electrons at temperature T, f the density of unoccupied recombination centres, H = the density of traps available, 1(T) = the TL intensity at temperature T, cs = the coefficient for thermal release of trapped charges, ~3= the coefficient for retrapping transition, y = the coefficient for recombination. These equations can be solved using an approximation that n is small compared to h and further that it remains constant during the TL process i.e. dn/dt 0 andf~h. This leads to a general solution for TL intensity: —
d.h dt
2et/T Ha R+(1 0b .-R)b’
with 1
b
1
R R
Ib\ \b 0/
1
~
b0
~
T
forR*0,
GB
/~
n ~ H
j~
A
Fig. 1. Simple model of TL process.
~E
(2a)
A. Sathyamoorthy et al.
/ c’omputer analysis of the thermoluminescence glow peaks 359
and
~ where R
~fe~t/TdT
=
~/y
=
forRO,
(2b)
the retrapping factor, a
0 = the frequency factor, b = h/H = occupancy, the constant heating rate, k = Boltzmann constant, F = trap depth, ~ = E/k. T0 = the temperature at which TL glow starts, b0 = the value of b at T = T0. The cases where R = 0 and R = 1 give the well-known first and second order kinetic equations of Randall and Wilkins [4] and Garlick and Gibson [5] respectively. These simple equations are good approximations to start with for glow curve analysis even for cases where more complex kinetics may govern the process. In fact, many subsequent theoretical improvements in TL theory were based on these simple cases in one form or another. Two important points about this model are to be noted: (i) The peak intensity is predicted at temperatures T10 where the followirit, relation holds between the various quantities. q
=
a0 bme_t/Tm ~~T~R+(l~
Reff,
(3)
where Reii
[ —
[2
—
(l—R)b01 R+(l R)b
may be called the effective kinetic order (note: limR~lReff = 2 and limR..+oReff The subscript m denotes the values of the quantities at T = Tm. (ii) Takingq dt = dT, the integration of first two terms of eq. (1) gives
h—h0
=
—(1/q)S(T),
where
S(fl=fI(fldT which can be rewritten as
b—b0=KS(T),
(4)
where the constant K depends on heating rate, H values and the light collection efficiency of the TL set up. This shows that the exact b’s versus fractional areas S(T)’s gives straight line.
360 A. Sathyamoorthy et al. / Computer analysis of the thermoluminescence glow peaks 3. Computational procedure Since the observed TL intensities are generally in arbitrary units, it is desirable to compare only the relative intensities I(T)/I(Tm) for both the observed and theoretical glow curves. In principle a computed curve which fits with the experimental one in the ‘least-square sense” can be selected by suitably modifying the parameters, such that the quantity
6
=
2
~ [J~XP(T) JC~P(T) 1~’~(T~) JCal(T)J 1
is a minimum. Instead of arbitrarily adjusting the parameters by iteration so as to satisfy this condition, we have considered the constraints imposed on the values of the parameters by eqs. (3) and (4). The procedure adopted for this purpose is described below. Taking the experimental value of q, T~’1= T~P= Tm and the approximate values of a 0, R and b0 non-linear eqs. (2) and (3) are solved simultaneously by iterative methods to get the consistent ~ and bm values. Using (2) the various b’s are then computed for the temperature range considered and the correlation coefficient p between these b’s and the corresponding fractional areas S(fl’s of the experimental curve is calculated (for a set of n values ofy f(x), p being defined as: 2 (x~— x5)(y1 — Ya)1 — — xa)2 ~(Y. — Ya)2
[~I~
whereya = ~iyiIn andxa = For a straight line the absolute value of p is I). The b
0 is then altered so as to increase p and the process is repeated till the change (~b0)in b0 between two consecutive cycles ~0.00l. The value of p was found to be of the order of 0.98 thereby showing a very good linear relationship between b’s and S(T)’s. Of the remaining parameters a0 and R, a0 was fixed and R changed to locate minimum in 6 by repeating the entire above mentioned process for eachR, till LXR ~0.00l. Finally, a0 was altered by a factor of 10 and the entire scheme repeated. The final set of a0, R, with the corresponding ~ value giving the least value of 6 was taken as the best set fitting the experimental glow curve. 4. Application This method of analysis has been applied to determine 60Co) the TLFe3~(0.0lMole%) parameters for 380 K glow peak observed in y-irradiated (dose 0.1 M rad doped MgO. The peak is well isolated except for a slight overlapping from the higher temperature peak. This peak was completely isolated from other peaks by the cleaning
A. Sathyamoorthy et a!.
/ Computer analysis of the ther,noluminesc~nceglow peaks 361
Table 1 Influence of initial trap occupancy (b
10
0) on TL peak Tm
0.1 0.3 0.5 0.7
380.0 K, R
1.0 o~j= 10
bm
~ K
6/n
p
0.0544 0.1622 0.2699 0.3773
8407.70 8805.05 8990.64 9112.90
0.02768976 0.02329216 0.02125374 0.02006609
0.986305 0.987829 0.988453 0.988855
Is.
technique recommended by Hoogenstraaten [6]. A computer fit was obtained using various sets of TL parameters. Table 1 shows the influence of initial trap occupany (b0) for one such set as measured by 6 and p. The consistent b1~and ~ values are also tabulated for each set. It is seen that either 6 and p can be used as a criterion for choosing the best fit. All the computational work was carried out on a BESM-6 computer. The parameter set giving the best fit (with p = 0.9996 13 and 6 = 0.040344) was: 10/s, R = 0.1004, E0.796eV. b0 = 1.0, a0 = l0 This method has also been applied to determine the TL parameters in case of natural calcite. That study and computer programme will be published as BARC report.
Acknowledgement Authors thank Dr. M.D. Karkhanavala for useful suggestions.
References [11 P. Braunlich, Thermoluminescence of geological materials, ed. D.J. McDougall (Academic Press, New York, 1968) p.61. [2J R. Chen, J. App!. Phys. 40 (1969) 570. [3] V. Maxia, S. Onnis and A. Rucci, J. Luminescence (1971) 378. [41 J.T. Randall and W.H.F. Wilkins, Proc. Roy. Soc. A 184 (1945) 366. [5] G.I.J. Garlick and A.F. Gibson, Proc. Phys. Soc. A 60 (1948) 574. [6] W. Hoogenstraaten, Philips Res. Rep. 13 (1958) 515.