Use of initial rise method to analyze a general-order kinetic thermoluminescence glow curve

Nuclear Instruments and Methods in Physics Research B 267 (2009) 3475–3479

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Use of initial rise method to analyze a general-order kinetic thermoluminescence glow curve N.S. Rawat a, M.S. Kulkarni a,*, D.R. Mishra a, B.C. Bhatt b, C.M. Sunta c, S.K. Gupta d, D.N. Sharma a a

Radiation Safety Systems Division, Bhabha Atomic Research Center, Trombay, Mumbai 400 085, India CSIR, Radiological Physics and Advisory Division, Bhabha Atomic Research Center, Trombay, Mumbai 400 085, India c C30/257, MIG Colony, Bandra (East), Mumbai 400 051, India d Technical Physics and Prototype Engineering Division, Bhabha Atomic Research Center, Trombay, Mumbai 400 085, India b

a r t i c l e

i n f o

Article history: Received 19 May 2009 Received in revised form 28 July 2009 Available online 6 August 2009 PACS: 47.54.Jk 78.60.Kn 87.16.A 94.05.Rx

a b s t r a c t The application of the initial rise method, in the case of the general-order kinetics of thermoluminescence, has so far been limited to finding the thermal activation energy. However, the order of kinetics and pre-exponential factors could not be evaluated using this method so one has to resort to other methods of glow curve analysis. In the present paper, a novel method is suggested to calculate the kinetic order and the pre-exponential factor from the Arrhenius plots of the initial rise part of the thermoluminescence glow curve. The method uses the intercept values on the thermoluminescence intensity axis of the Arrhenius plots at two or more known doses to evaluate the value of kinetic order. Ó 2009 Elsevier B.V. All rights reserved.

Keywords: Initial rise method Kinetic order Pre-exponential factor Frequency factor

1. Introduction

assuming the frequency factor remains same and there is no overlapping of glow peaks, i.e.

The thermoluminescence (TL) characteristic of any material is defined by parameters such as kinetics, trap depth E and frequency factor s. Since the pioneering work of Randall and Wilkins [1] and Garlick and Gibson [2] there has been a plethora of published papers dealing with methods by which the trapping parameters (mainly E and s) can be obtained from a TL glow curve. The glow peak temperature Tm is independent of radiation dose in first order kinetics whereas in second order kinetics, the glow peak temperature Tm decreases with increasing dose, and the TL glow curves representing second order kinetics are more symmetric than that of the first order TL glow curves. The initial rise method is the simplest experimental procedure that was first suggested by Garlick and Gibson [2] to obtain the trap depth and is independent of the kinetics involved. In the initial rise part of a TL curve (early rising range of temperatures), i.e. T  Tm (where Tm is the temperature at which the TL maxima occurs), the rate of change of trapped carrier population is negligible. Hence the TL intensity I is strictly proportional to exp(E/kT),

  E IðTÞ ¼ A  exp  kT

* Corresponding author. Tel.: +91 22 25595076; fax: +91 22 25505151. E-mail address: [email protected] (M.S. Kulkarni). 0168-583X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2009.08.002

where A is a constant, I(T) is the TL intensity at any temperature T, when the sample is heated at a linear heating rate of b = dT/dt, E is the thermal activation energy and k is Boltzmann’s constant. If the plot of ln(I) vs. 1/T is made over this initial rise region (T  Tm), then a straight line of slope E/k is obtained from which the activation energy E is easily found. The important requirement in this analysis is that n (the concentration of the trapped carriers at any instant) remains approximately constant. Only upon increase in temperature beyond a critical value, Tc, does this assumption become invalid. However, the initial rise technique can be used only when the glow peak is well defined and clearly separated from the other peaks. Halperin and Braner [3] suggested an improvement to the initial rise method considering general-order kinetics, i.e.

  E IðTÞ ¼ s0 nb exp  kT

ð1Þ

where b is the order of kinetics and s0 is the pre-exponential factor. If b is the constant heating rate, n can be calculated

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R1 R1 as: t Idt ¼ 1b T IdT and if b is known a plot of lnðI=nb Þ vs. 1/T yields a straight line with slope E/k. The general-order kinetics expression as proposed by May and Partridge [4] is:

IðTÞ ¼ b

  dn E ¼ Cs0 nb exp  dT kT

In this paper, we are proposing a new method for the determination of TL parameters including kinetic order, pre-exponential factor and frequency factor using the initial rise method for TL equation of general-order kinetics over a varying set of doses.

ð2Þ 2. Determining kinetic order

where I is expressed in the units of cm3 s1 and n in cm3, C is a scaling factor, the quantity s0 turns out to have the dimensions cm3(b1) s1. In order to balance the dimensions in Eq. (1), Rasheedy [5] proposed that s0 may be replaced by s0 = s/Nb1. Here, s is frequency factor (s1) and N the total concentration of the traps (cm3). This definition of s0 fits well into the first and second order kinetics expressions of Randall and Wilkins [1] and Garlick and Gibson [2] in which b = 1 and b = 2, respectively and the corresponding s0 values are equal to s and s/N, respectively. Sunta et al. [6] have used the successive interpolation method to find the values of s0 corresponding to the range of b values from b = 1 to b = 2. They arrived at the same expression as above stated expression of Rasheedy. Eq. (2), thus may be written as:

IðTÞ ¼ Cs

Nb1

E kT

ð4Þ

where n0 is the initial concentration of trapped electrons. The first term on the right hand side is the intercept on the ln IðTÞ vs. 1/T straight line. Call it I1 . Thus I1 ¼ ln Cs0 nb0 . Now if the dose to the sample is increased by a factor X, and the initial rise curve is recorded with the same b value, the intercept becomes ln Cs0 ðXn0 Þb , called I2 . It may be seen that

I2  I1 ¼ ln Cs0 ðXn0 Þb  ln Cs0 nb0

  E exp  kT

ð3Þ

I2  I1 ¼ lnðXÞb

15

6.0x10

9

15

5.5x10

15

9

5.0x10

0

n0=10

4.5x10 4.0x10

4

9

15

3.5x10

15

10 15 10 17 10

2

9

15

(b) 5.0x10

12

13

15

TL Intensity (arb. units)

ln IðTÞ ¼ ln Cs0 nb0 

n0/N=10

1

4.5x10

10

15

5

15

3.0x10

15

2.5x10

15

2.0x10

15

1.5x10

-5

-4

15

TL Intensity (arb. units)

(a)

nb

In the initial rise part when the condition n = n0 applies, we have from Eq. (2)

4.0x10

10

15

3.5x10

10 -2 10 1

2

4

10

5

15

3.0x10

15

2.5x10

15

2.0x10

15

1.5x10

15

15

1.0x10

14

5.0x10

1.0x10

14

5.0x10

0.0

0.0

325

350

375

400

425

450

300

350

400

450

500

550

600

Temperature (K)

Temperature (K) 15

(c) 4.5x10

15

4.0x10

n0/N=10

1

10 -2 10 1

TL Intensity (arb. units)

15

3.5x10

10

15

3.0x10

-5

-4

2

15

2.5x10

10

15

2.0x10

4

10

15

5

1.5x10

15

1.0x10

14

5.0x10

0.0 400

600

800

Temperature (K) Fig. 1. Numerically simulated TL curves with input parameters as E = 1 eV, s = 1012 s1, N = 1017 cm3, n0 = 1012,. . .,1017. The respective TL curves are multiplied with a factor shown. (a) b = 1, (b) b = 1.5 and (c) b = 2.

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(a) 34

n0=10

32

(b) 34

12

32

13

10 15 10 17 10

30 28

30 28 26 24

ln TL

ln TL

26 24

22 20 18

22

16

20

14

18

-5

n0/N=10

12

16

10

14

8

-4

10 -2 10 1

6

12 0.0028

0.0030

0.0032

0.0034

0.0022

0.0036

0.0024

0.0026

0.0028

0.0030

0.0032

0.0034

0.0036

-1

-1

1/T (K )

1/T (K )

(c) 35 30

ln TL

25 20 15 10

n0/N=10

-5

-4

5

10 -2 10 1

0 0.0021

0.0024

0.0027

0.0030

0.0033

0.0036

-1

1/T (K ) Fig. 2. Initial rise plots for curves in Fig. 1, here the points considered are up to the 5% of TL peak intensity. (a) b = 1, (b) b = 1.5 and (c) b = 2.

350 300

Dose=100mGy 300mGy 500mGy

1.6 1.4 1.2 1.0

250

0.8

200

ln ITL

TL Intensity (arb. units)

1.8

Dose=100mGy 300mGy 500mGy

0.6 0.4

150

0.2

100

-0.2

0.0 -0.4 -0.6

50

-0.8 -1.0

0 100

200

300

400

0

Temperature ( C) Fig. 3. Recorded 230 °C TL glow peak of CaSO4: Dy samples for 4 K/s heating rate.

0.00249

0.00252

0.00255

0.00258

-1

1/T (K ) Fig. 4. Initial rise plots of 230 °C TL peak of CaSO4: Dy.

0.00261

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We thus have

b¼

4. Determining frequency factor

I2  I 1 ln X

ð5Þ

The intercepts I1 and I2 being measurable at two doses differing by a factor of X, the b value may be readily found. Here it is assumed that the traps fill linearly with dose and the total TL intensity also grows linearly with trap occupancy.

3. Determining pre-exponential factor

I ¼ ln Cs0 nb0 ¼ ln s0 þ ln Cnb0 or I ¼ ln s þ ln Cn0 þ

I ¼ ln Cs0 nb0 ¼ ln Csn0

n b1 0

N

This leads to:

Again from the intercept of the ln IðTÞ vs. 1/T graph of the initial rise part, the intercept I may be expressed as (see Eq. (1))

0

The frequency factor s is a meaningful quantity as opposed to the empirical quantity s0 . To find the s value, one may use the Eq. (3) to plot the ln IðTÞ vs. 1/T graph for the initial rise part. The intercept I now is given by

ln n0b1

h n i 0 s ¼ Anti log I  ln Cn0  ðb  1Þ ln N h n i 0 or s ¼ Anti log I  ln A  ðb  1Þ ln N

ð7Þ

ð6Þ

or s0 ¼ Anti log ½I  ln Cn0  ðb  1Þ ln n0  Having measured the b value from Eq. (4), the value of s0 may be found using the fact that Cn0 = total area of the glow peak. One may be able to find the n0 value from the optical absorption band if such a band could be found for the concerned TL traps. In the present work the input value of n0 is used to simulate the results.

This equation also requires knowledge of the area under the TL peak (Cn0 ¼ A). Additionally n0/N also is required. The latter may be estimated from the TL intensity vs. dose curve. If one assumes saturating exponential pattern for the TL intensity growth, one has at n0/N = 1, the TL intensity is nearly saturated. Actually at n0/ N  0.632 [7]. One may estimate n0/N at lower doses assuming linear relation.

Table 1 Calculated values of E, b and s for different trap occupancies for input b = 1. Trap occupancy or dose (D)

Intercept (I)

TL area (Cn0)

Dose ratio (X*)

Trap depth, E (eV)

Order of kinetic, b**

Frequency factor, s***

1012 (D1) 1013 1015 1017

55.2 (I1) 57.5 62.1 66.7

1012 1013 1015 1017

1 10 103 105

0.998 0.998 0.998 0.998

– 1.0 1.0 1.0

0.936  1012 0.936  1012 0.936  1012 0.936  1012

*

X = D/D1. b ¼ I2lnIX1 . *** s ¼ Anti log½I  ln Cn0 . **

Table 2 Calculated values of E, b, s and s0 for different trap occupancies for input b = 1.5. Trap occupancy (D) 5

10 (D1) 104 102 1 * **

Intercept (I) 49.5 (I1) 52.9 59.8 66.7

TL area (Cn0) 12

10 1013 1015 1017

Dose ratio (X) 1 10 103 105

Trap depth, E (eV) 0.999 0.998 0.998 0.997

Order of kinetic, b – 1.497 1.497 1.495

Frequency factor, s* 12

0.964  10 0.935  1012 0.932  1012 0.920  1012

Pre-exponential factor, s0 ** 3260 3260 3307 3407

  s ¼ Anti log I  ln Cn0  ðb  1Þ lnðnN0 Þ . 0 s ¼ Anti log ½I  ln Cn0  ðb  1Þ ln n0 .

Table 3 Calculated values of E, b, s and s0 for different trap occupancies for input b = 2. Trap occupancy (D) 5

10 (D1) 104 102 1 *

Intercept (I) 43.7 (I1) 48.3 57.5 66.7

TL area (Cn0) 12

10 1013 1015 1017

Dose ratio (X) 1 10 103 105

Trap depth, E (eV) 0.998 0.998 0.998 0.998

Order of kinetic, b – 1.997 1.996 1.993

Frequency factor, s* 12

0.977  10 0.971  1012 0.953  1012 0.909  1012

Pre-exponential factor, s0 * 1.044  105 1.044  105 1.078  105 1.161  105

Expressions for s and s0 are same as that used in Table 2.

Table 4 Experimentally calculated TL parameters for 230 °C TL peak of CaSO4: Dy.

*

Dose (D) (mGy)

Intercept (error) (I)

TL area (Cn0)

Dose ratio (X)

Trap depth, E (eV)

Order of kinetic, b

Frequency factor, s*

100 (D1) 300 500

31.04 (0.21) (I1) 32.28 (0.14) 32.96 (0.14)

2038 6190 9130

1 3 5

1.13 1.14 1.14

– 1.13 1.19

1.49  1010 1.69  1010 2.26  1010

s ¼ Anti log½I  ln Cn0 .

N.S. Rawat et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 3475–3479

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However, for first order kinetic TL process Eq. (7) on substituting b = 1 reduces to the expression

7. Results and discussion

s ¼ Anti log½I  ln A

The agreement between the values found from simulation and the input values in the case of E, b and s is quite good. The values found for these quantities are almost independent of changes in trap occupancy n0. On the other hand the value found for s0 has a strong dependence on b. This is not unexpected, since as discussed above the dimension of s0 are cm3(b1) s1, which indicates that s0 would undergo changes as b changes. The values found for s0 listed in Tables 1–3 are in agreement with the definition s0 = s/Nb1, which support the validity of Eq. (6). The experimental values of E, b and s as calculated for 230 °C TL peak are in good agreement with the values reported in the literature. To sum up, the intercepts of the Arrhenius plots of initial rise part of TL glow peaks can be used to find b, s or s0 . The expressions derived for finding these parameters for a glow peak are general in nature. As may be seen from the results presented in Tables 1–3 these expressions are applicable to the glow peaks of any order of kinetics, irrespective of whether the said peaks are of first, second or general order.

ð8Þ

Clearly, determining the frequency factor s, from Eq. (8) for first order kinetic process is very simple and it requires intercept value of initial rise plot and area under the TL peak, both of which can be experimentally obtained. 5. Simulation The initial rise intensities IðTÞ were computed numerically using the following input parameters in Eq. (3): E = 1 eV, s = 1012 s1, C = 1, N = 1017cm3 and b = 2 K/s. Fig. 1 shows numerically simulated TL glow curves for using b = 1, 1.5 and 2. Computations were repeated at various n0 values. The Runge–Kutta second order technique is used. Initial rise IðTÞ values were calculated until IðTÞ value reached no more than 5% of the peak intensity as shown in Fig. 2. Intercepts of the Arrhenius plots were calculated by fitting the straight line equation to the computed ln IðTÞ values at different T.

8. Conclusions 6. Experimental It is difficult to find a phosphor material that has clean and well resolved TL peaks. However, using the technique of thermal cleaning one can remove the satellite or embedded peaks, if any, in the TL glow curve of a phosphor material. For the experimental validation of the proposed technique, CaSO4: Dy phosphor samples weighing 5 mg each were taken and exposed to 100 mGy, 300 mGy and 500 mGy absorbed dose of 90Sr/90Y beta source. The analysis of the high temperature TL peak at 230 °C is performed. The samples were given thermal treatment up to 150 °C to remove low temperature TL peaks. The TL glow curve peaking at 230 °C is recorded at 4 K/s heating rate as shown in Fig. 3. Fig. 4 shows the initial rise plot of TL glow curves (as shown in Fig. 3). Table 4 shows the analysis of 230 °C TL glow peak of CaSO4: Dy. The order of kinetics is found to be 1.13 and 1.19. For the sake of simplicity and calculation of frequency factor we have assumed that first order kinetics is followed by 230 °C TL peak of CaSO4: Dy. The frequency factor at three known doses is of the order 1010 s1 as shown in Table 4. The experimentally calculated TL parameters using the proposed technique (Table 4) is found to match with the literature [8].

It has been shown that the initial rise method can be used to determine the order of kinetics, pre-exponential factor and frequency factor apart from trap depth E. Using theoretical simulations and finding the trapping parameters derived equations we have shown its validity. The essence of this technique lies in the fact that all these parameters can be determined independently, i.e. the expression for the determination of b and s parameters is independent of E. Thus the uncertainty and error associated with one parameter does not propagate into the determination of another. However, all the limitations associated with initial rise method are applicable here too. References [1] [2] [3] [4] [5] [6]

J.T. Randall, M.H.F. Wilkins, Proc. R. Soc. A 184 (1945) 366. G.F.J. Garlick, A.F. Gibson, Proc. Phys. Soc. 60 (1948) 574. A. Halperin, A. Braner, Phys. Rev. 117 (1960) 408. C.E. May, J.A. Partridge, J. Chem. Soc. 40 (1964) 1401. M.S. Rasheedy, J. Phys.: Condens. Matter 5 (1993) 633. C.M. Sunta, B.C. Bhatt, P.S. Page, in: Proceedings 3rd International Conference on Luminescence and Its Applications, New Delhi, 13–16 February 2008, p. 30. [7] C.M. Sunta, E.M. Yoshimura, E. Okuno, Radiat. Meas. 23 (1994) 655. [8] Numan Salah, P.D. Sahare, S.P. Lochab, Pratik Kumar, Radiat. Meas. 41 (2006) 40.