computerised glow curve deconvolution using ...

Radiation Protection Dosimetry Vol. 101, Nos. 1–4, pp. 47–52 (2002) Nuclear Technology Publishing

COMPUTERISED GLOW CURVE DECONVOLUTION USING GENERAL AND MIXED ORDER KINETICS

Abstract — Some accurate glow curve fitting functions for general and mixed order kinetics glow peaks are proposed and discussed. These mathematical expressions are used together with peak search and non-linear minimisation algorithms in order to provide a fast glow curve deconvolution for those materials which cannot be well fitted using first order kinetics. To test the accuracy of the proposed method, the result of the fitting of synthetic glow curves is compared with the original data giving negligible errors for values of parameters currently found in TL materials.

INTRODUCTION

KINETIC EQUATIONS

Computerised glow curve deconvolution used in thermoluminescence (TL) dosimetry and based on fitting methods depends on the shape assumed for single glow peaks, i.e. on the mathematical equations provided by TL models (1,2). The use of some synthetic dosemeters as well as some natural materials giving glow curves which cannot be well fitted by 1st order kinetics made necessary the use of new peak shape functions based on more complex descriptions of TL processes. Unfortunately, most of these descriptions do not yield explicit expressions suitable for fitting algorithms. Although there is not an agreement about what model should be preferred, the generally good fits of many experimental glow curves by the so called general and mixed order kinetics makes them good candidates to be considered as simplified descriptions of TL mechanisms for glow curve analysis in TL dosimetry (1). Iterative fitting methods require many evaluations of the fitting functions. Therefore, simple explicit equations give faster algorithms, especially if a good first estimation of the parameters can be provided. In this paper, some explicit equations for general and mixed order kinetics in the form I(T,TM,IM,E,b) and I(T,TM,IM,E, ␣) respectively were discussed (3,4). They have two free parameters, TM and IM, directly estimated from measured glow curves instead of the kinetics parameters, s and n0, whose values are not initially known. A glow curve analysis software has been developed based on such general and mixed order kinetics models. The program automatically finds and fits up to 10 overlapping glow peaks enabling the main kinetic parameters (s, E, b, ␣) to be estimated for each individual peak together with its position, maximum intensity and individual areas in a very short time.

General order kinetics The solution of the general order kinetics equation for a constant linear rate is (5):

冉冊再冕冉冊冎 E kT s 1 +(b ⫺ 1) ␤

I(T) ⫽ sn0 exp ⫺

T

E exp ⫺ dT⬘ kT⬘ T0

(1)

⫺b

/(b⫺1)

where n0 is the initial density of trapped charge carriers, s is the frequency factor, k is the Boltzman constant, E is the activation energy, b is the kinetic order (1ⱕbⱕ2) and T⫽T0+␤t is the linear heating profile. The temperature of the maximum, TM is obtained by differentiating and setting dI(t)/dt⫽0, giving (6):

再

E s 1 + (b ⫺ 1) kT2M ␤

冉冊

冕冉冊冎 TM

T0

E exp ⫺ dT⬘ kT⬘

(2)

E s ⫽ b exp ⫺ ␤ kTM

Then IM⫽I(TM). This relationship between the parameters can be used to rewrite Equation 1 as an explicit function of (E, b, TM, IM): I(T) ⫽ IM exp

冉

exp

冉

冊再冉冊冕冊冎

E b⫺1 E E ⫺ 1+ kTM kT b kT2M

E E ⫺ dT⬘ kTM kT⬘

T

(3)

TM

⫺b (b⫺1)

/

The integral in Equations 1 and 3 cannot be solved in terms of elementary functions (3,5). One of the simplest possible approximations to evaluate Equation 3 is to consider: exp and

Contact author E-mail: jm.gomezros얀ciemat.es

47

冉

冊冋

E E E ⫺ ⬇ exp (T ⫺ TM) kTM kT kT2M

册

(4)

Downloaded from http://rpd.oxfordjournals.org/ at National Institute for Physics and Nuclear Engineering IFIN-HH on July 11, 2012

J. M. Go´mez Ros† and G. Kitis‡ †CIEMAT. Av. Complutense 22. 28040 Madrid, Spain ‡Aristotle University of Thessaloniki. Nuclear Physics Laboratory. 54006 Thessaloniki, Greece

E kT2M

冕冉冋 T

´ MEZ ROS and G. KITIS J. M. GO

冊

E M ⫺ dT⬘ kTM kT⬘

exp

TM

册

for z⬎1. So, the error is negligible for temperatures (in K) T⬍1.16 ⫻ 104E, where E is the activation energy (in eV) (9). Then, finally, Equation 10 is obtained:

(5)

冉冊再冉冊冋冉冊冉冊冉冊冉冊册冎

E (T ⫺TM) ⫺ 1 kT2M

⬇ exp

E E b⫺1 E ⫺ 1+ kTM kT b kTM T E E E exp ⫺ F TM kTM kT kT ⫺b b⫺1 E /( ) ⫺F kTM

I(T) ⫽ IM exp

Thus we have:

册再冉冊

冋

册冎

(6)

As in the previous simple approximation, the area of the glow peak can be calculated as:

The kinetic parameters appearing in Equation 1 can be obtained from the values of TM and IM using the maximum condition of Equation 2. In particular, the area of the glow peak will be: Area ⬀ n0 ⫽ ⫺

冕

⬁

n(t)dt ⫽ ⫺

0

⬇

2 M b (b⫺1)

kIMT b/ ␤E

1 ␤

冕

Area ⬀ n0 ⫽

⬁

n(t)dt

T

E E ⫺ dT⬘ kTM kT⬘

exp

TM

⫽ T exp

(7)

The solution of the general order equation for a constant heating rate is give by (5): I(T) ⫽

冉冊冉冊

1 + RM

E exp ⫺ dT⬘ kT⬘ T0

T

E exp ⫺ dT⬘ ⫺␣ kT⬘ T0

冉冊

cs⬘ E E R exp ⫽ kT2M M kTM ␤ where RM is defined as:

2

冋冉冊冋冉冊

(13)

冉冊册冉冊册

cs⬘ T E 兰 M exp ⫺ dT⬘ ⫺␣ ␤ T0 kT⬘ RM = cs⬘ T E exp 兰 exp ⫺ dT⬘ +␣ ␤ T0 kT⬘ exp

(9)

(14)

It is not possible to obtain an explicit expression for Equation 12 depending on TM, IM, as was done for general order kinetics (4). If the maximum condition of expressions 13-14 is used and the rational approximation 9, then Equation 15 is obtained:

b0 ⫽ 1.681534 b1 ⫽ 3.330657

冉

冊再冋冉冊冉冊冉冊冉冊册冎冊再冋冉冊冉冊冉冊冉冊册冎冉冊冎 E E T E E E E E ⫺ exp RM exp ⫺ F ⫺F kTM kT kTM TM kTM kT kT kTM E T E E E E exp RM exp ⫺ F ⫺F ⫺ 1 ⫺ RM kTM TM kTM kT kT kTM

4R2M exp

再冉

cs⬘ ␤

T

where ␣⫽n0/(n0+c). The condition for the maximum can be written as (4,6):

(8)

where an upper bound for the error is 兩⑀ (z)兩 ⬍5 ⫻ 10⫺5

I(T)⫽ IM

冉冊冋冉冊冕冉冊册再冋冉冊冕冉冊册冎 exp

F(z) ⫽ ez E2(z) ⫽ 1 ⫺ zez E1(z)

a0 ⫽0.250621 a1 ⫽2.334733

(12)

cs⬘ ␤

E exp s⬘ c ␣ exp ⫺ kT 2

where E2(x) is the exponential integral of second order (7,8). Now, a very accurate rational approximation can be obtained if the following equation is defined (7):

a0 + a1 z + z2 + ⑀(z) b0 + b1 z +z2

(11)

Mixed order kinetics

E E E E E ⫺TM exp E kTM 2 kT kTM 2 kTM

⫽1⫺

再冉冊冉冊冉冊册冎

b⫺1 E bkIMTM 1⫺ ␤E b kTM ⫺b b⫺1 E /( ) E F exp ⫺ kTM kTM

0

Equation 5 is not defined for b⫽1 but the limit exists and this equation becomes the Podgorsak approximation for 1st order kinetics (1,5) when b approaches to 1. Although Equation 6 may be accurate enough for some practical purposes (see the discussion in the last section), some better methods can be used to approximate the integral (1,3). Furthermore, the integral can be expressed as:

冊冕冉冉冊冉冊

(10)

48

2

(15)


冋

E 1 b⫺1 + (T ⫺ TM) kT2M b b ⫺b b ⫺1 E /( ) (T ⫺ TM) exp kT2M

I(T) ⫽ IM exp

GLOW CURVE DECONVOLUTION

Equations 13 and 14 can be combined, giving:

冉

冊

1 + RM ␣ 1 ⫺ RM

冋

⫽ exp RM

The glow peak area is: Area ⫽

(16)

冉冊冉冊册

E E E exp E kTM kTM 2 kTM

The implementation of the fitting functions discussed in the previous section is quite straightforward but there are some details that should be mentioned. Glow curve fitting is done using an iterative Levenberg–Marquard algorithm and a finite difference Jacobian to minimise the chi-squared function ␹2 defined as:

(17)

␹2 ⫽ RM (α,10) RM (α,100) RM (approx.)

0.4 0.2 0 0.4

0.6

0.8

n⫽1

冊

2

In(Ti)

(19)

where N is the number of peaks, I(Ti) and In(Ti) are the measured glow curve and the nth fitting glow peak for temperature Ti respectively (9). Glow curve fitting algorithms require an initial estimate of the peak parameters sufficiently accurate to guarantee the convergence and to minimise the number of iterations. The new equations have the main advantage compared with the original ones that they depend on a set of parameters, (TM, IM, E, b) or (TM, IM, E, ␣), which can be directly estimated from the measured glow curve for a single isolated glow peak considering the temperature and intensity of the maximum and the peak shape and symmetry (i.e. the left and right partial width at half the maximum intensity). In the case of several overlapping peaks the parameters E and b (or ␣ in the case of mixed order

0.6

0.2

冘 N

I(Ti)⫺

i

1.0

RM (α,w)

冘冉

1.0

α

Figure 1. Behaviour of RM(␣,w) as a function of ␣ for different values of w⫽E/kTM.

Table 1. Kinetics parameters used in a synthetic glow curve and results of the fitting with Equations 6 and 10. Deviations (in%) are indicated. Parameter

Peak 1

Peak 2

peak 3

s no E b TM IM s no E b TM IM s no E b TM IM FOM

Value

Simple approximation

1.00 ⫻ 1013 5.00 ⫻ 104 1.050 1.200 394.55 6950.8 5.00 ⫻ 1017 1.00 ⫻ 105 1.500 1.900 419.48 13333.9 1.00 ⫻ 1013 2.00 ⫻ 104 1.500 1.500 521.16 2005.0

− 49851.89 1.19 1.35 394.50 6950.7 – 100225.27 1.61 1.99 419.50 13342.1 – 19844.24 1.71 1.69 521.15 2005.2 0.82%

49

Rational approximation − (⫺ 0.30%) (13.33%) (12.50%) (⫺ 0.01%) (0.00%) – (0.23%) (7.33%) (4.74%) (0.00%) (0.06%) – (⫺ 0.78%) (14.00%) (12.67%) (0.00%) (0.01%)

− 50005.40 1.049 1.201 394.61 6954.28 – 99991.80 1.502 1.899 419.42 13340.57 – 20000.80 1.500 1.501 521.14 2006.00 0.07%

⫺ (0.01%) (⫺ 0.10%) (0.08%) (0.02%) (0.05%) – (⫺ 0.01%) (0.13%) (⫺ 0.05%) (⫺ 0.01%) (0.05%) – (⬍ 0.01%) (⬍ 0.01%) (0.07%) (⬍ 0.01%) (0.05%)


RM ⫽ (1 ⫺ ␣)(1 + 0.2922␣ ⫺ 0.2783␣2)

0

(18)


which is a definition of RM as an implicit function of the two parameters ␣ and w⫽E/kTM (4). It is immediately obvious that RM(0,w)⫽1 and RM(1,w)⫽0. For values of w between 10 and 100, the dependence of RM on w can be neglected (Figure 1) and it can be evaluated with a very good approximation as a function of ␣ as:

0.8

4kIM T2M ␣RM ␤⌬tE (1 ⫺ ␣)(1 ⫺ R2M)

´ MEZ ROS and G. KITIS J. M. GO

to the other parameters (1,9), so the first estimation of E and b (or ␣) is not critical for the convergence if the number of peaks and the initial values for TM are well estimated.

kinetics) cannot be estimated from the glow curve. Moreover, it is necessary to determine the number of peaks to be fitted. Nevertheless, the ␹2 function is much more dependent on the position of the peaks (TM) than 16

(a) Downloaded from http://rpd.oxfordjournals.org/ at National Institute for Physics and Nuclear Engineering IFIN-HH on July 11, 2012

14

12

Intensity (x103)

10

8

6

4

2

0 2

(b)

Residuals (x102)

1 0 -1 -2 300

350

400

450

500

550

600

Temperature (K)

Figure 2. Glow curve fitting (a) and residue (b) of a synthetic glow curve assuming general order kinetics and two fitting functions: simple approximation (dashed line) and rational approximation (solid line), described in the text. 50


For glow curves with low noise/signal ratio, the local minima of the smoothed second derivative are used to detect the position of the peaks and the maximum intensity. In high noise curves the combined analysis of the

smoothed glow curves together with the 1st and 2nd derivatives can still be used to reveal hidden peaks once the number of peaks has been provided and some criteria were established to reject singularities due to the

(a) Downloaded from http://rpd.oxfordjournals.org/ at National Institute for Physics and Nuclear Engineering IFIN-HH on July 11, 2012

100

80

60

Intensity

40

a = 1.0

20 a=0

0

Residuals

0.2

(b)

0.1 0

-0.1 -0.2 350

400

450 Temperature (K)

500

550

Figure 3. Glow curve fitting (a) and residue (b) of a simulated mixed order kinetics glow peak with TM⫽450 K, IM⫽100, E⫽1.5 eV and different values of ␣ (0.0, 0.2, 0.4, 0.6, 0.8 and 1.0). 51

´ MEZ ROS and G. KITIS J. M. GO (2,9)

racy was drastically improved when Equation 10 was used. An example of the accuracy of Equation 15 as a fitting function for mixed order kinetics is depicted in Figure 3 for synthetic glow peaks calculated for different values of ␣. In all the cases the deviation on the evaluated parameters is less than 0.01%. The main advantage in the use of Equations 6, 10 or 15 for glow curve fitting is that all the parameters involved have a geometrical meaning so they can be estimated from the shape of the glow peak in experimental glow curves. Furthermore, the proposed equations include only elementary functions and algebraic operations so they have an optimal computational performance, especially when iterative minimisation algorithms are considered. The accuracy depends only on the method used to evaluate the exponential integral and the results are very good when a rational approximation is used. The kinetic parameters appearing in the original equations (s and n0) and individual peak areas are obtained from the fitted ones throughout the maximum condition relationship. These fitting functions together with the possibility of using a peak search algorithm to get a first estimation of the parameters in each measured glow curve have been used to develop an automated glow curve deconvolution program for general and mixed order kinetics. Computing time required by the program in Pentium III based computers is just about few seconds for each curve. The program has been employed to study the glow curve structure and dose dependence of individual glow peaks in a synthetic quartz (10).

RESULTS AND CONCLUSIONS Figure 2 illustrates the results of using Equations 6 (dashed line) and 10 (solid line) to fit a synthetic glow curve. Each peak was computed by solving the general order differential equation (8) with the kinetic parameters listed in Table 1. A linear heating profile with a constant heating rate ␤⫽5 K.s⫺1 and an initial temperature T0⫽300 K was employed. The fitted parameters compared with the actual ones are shown in Table 1. The values obtained for E and b with the simple approximation 6 are very poor but this equation gives a good estimation of the peak areas (less than 1% in deviation) and a good fitting of the whole glow curve. The accuREFERENCES

1. Horowitz, Y. S. and Yossian, D. Computerised Glow Curve Deconvolution: Application to Thermoluminescence Dosimetry. Radiat. Prot. Dosim 60 (1995). 2. Delgado, A. and Go´ mez Ros, J. M. Computerised Glow Curve Analysis: A Tool for Routine Thermoluminescence Dosimetry. Radiat. Prot. Dosim. 96(1–3), 127–132 (2001). 3. Kitis, G., Go´ mez Ros, J. M. and Tuyn, J. W. N. Thermoluminescence Glow Curve Deconvolution Functions for First, Second and General Orders of Kinetics. J. Phys. D: Appl. Phys. 31, 2636–2641 (1998). 4. Kitis, G. and Go´ mez Ros, J.M. Thermoluminescence Glow Curve Deconvolution Functions for Mixed Orders of Kinetics and Continuous Trap Distribution. Nucl. Instrum. Methods A 440, 224–231 (2000). 5. Chen, R. and Kirsh, Y. Analysis of Thermally Stimulated Processes (Oxford: Pergamon Press) (1981). 6. Yossian, D. and Horowitz, Y. S Mixed-Order and General-Order Kinetics Applied to Synthetic Glow Peaks and to Peak 5 in LiF:Mg,Ti (TLD–100). Radiat. Meas. 27, 465–471 (1997). 7. Abramowitz, M. and Stegun, I. A. Handbook of Mathematical Functions (New York: Dover) (1972). 8. Hoogenboom, J. E., de Vries, W., Dielhof, J. B. and Bos, A. J. J. Computerized Analysis of Glow Curves from Thermally Activated Processes. J. Appl. Phys. 64, 3193–3200 (1988). 9. Go´ mez Ros, J. M., Sa´ ez Vergara, J. C., Romero, A. and Budzanowski, M. Fast Automatic Glow Curve Deconvolution of LiF:Mg,Cu,P Curves and its Application in Routine Dosimetry. Radiat. Prot. Dosim. 85, 249–252 (1999). 10. Go´ mez Ros, J. M., Correcher, V., Garci´a Guinea, J. and Delgado, A. Kinetic Parameters of Lithium and Aluminium Doped Quartz from Thermoluminescence Glow Curves. Radiat. Prot. Dosim. 100(1–4), 399–402 (2002).

52


noise . Essentially, true peaks correspond to those local minima of the second derivative for which two additional requirements are fulfilled: (i) the smoothed 1st derivative is a decreasing function in some certain neighbourhood of the considered point and (ii) the smoothed glow curve intensity is greater than some threshold value (9). Glow curve deconvolution requires a minimum signal to noise ratio to produce reliable results. Otherwise, convergence is not achieved, it terminates at a non-critical point or it gives absurd values for the fitting parameters. In general, it can be used for doses higher than approximately 10 times the minimum measurable dose for a given TL material. For lower doses, other glow curve analysis methods seems to be more appropriated (2).