J. PRAKASH and D. PRASAD: The Appearance of a Thermoluminescence Glow Curve 28 1

phys. stat. sol. (a) 142, 281 (1994)

Subject classification: 78.60

Depurtment of Physics, Uniaersitj of Goraklipur ')

Mechanisms Responsible for the Appearance of a Thermoluminescence Glow Curve

BY J. PRAKASH and D. PRASAD

The thermoluminescence (TL) intensity of first- and second-order glow curves is found to be devoid of any correlationship. The TL intensities of second- and higher-order kinetics are, howevey, representable by a general equation. This discrepancy initiated a reconsideration of the existing mechanisms responsible for the appearance of a TL glow curve. In such an attempt the Adirovitch set of equations is modified. A theoretical model is built up which successfully explains the TL intensity of various orders of kinetics including first order. It is found that irrespective of the order of the kinetics involved, TL intensity decays exponentially.

1. Introduction

Whenever an insulator or a semiconductor is irradiated, electrons are excited from the valence band to the conduction band. Most of the excited electrons return to the valence band after a very short time (= lo -* s) emitting a photon. The emission of photon thus obtained is known as fluorescence. If the excited electron instead of returning back to the valence band goes to some metastable state or is trapped in some trap level present in the forbidden gap, the electron can stay there for a longer time. Once the electron is trapped, it needs energy to be raised to the conduction band and only then can return to the valence band. As soon as it gets the required amount of energy it shows a delayed luminescence known as phosphorescence. If the required energy is supplied thermally, the phenomenon is called thermoluminescence (TL). The occurrence of the TL glow curve, i.e. variation of the TL intensity with the temperature is assigned to the processes which are either recombination dominant [ 11 or where retrapping and recombination probabilities are equal [2]. Consequently, a TL glow curve involving either a first-(monomolecular) or a second- order kinetics (bimolecular) is observed. In the first-order kinetics, the intensity of a TL glow curve is expressed as [3]

where no represents the initial concentration of trapped electrons per unit volume at t = 0, s the escape frequency factor or pre-exponential factor, E , the trap depth or activation energy, k the Boltzmann constant, T the absolute temperature, h the linear heating rate, and To the initial temperature at which the TL glow curve starts to appear. In the glow

I ) Gorakhpur 273009, India.

282 J. PRAKASH and D. PRASAD

curve maximum TL intensity appears at a temperature TM such that

It has been observed that the peak position is unaltered by a change in no but it changes by a change in b. The peak position is found to shift [4] to higher temperatures with increasing value of b.

In systems involving second-order kinetics, where recombination and retrapping pro- babilities are equal [2, 31, the TL intensity is represented by

where N represents the total concentration of electron traps per unit volume. It should be mentioned that in developing (3) it has been assumed [3] that the traps are far from saturation (i.e. n N) .

It has been observed that in second-order kinetics TM changes due to a change in b similar to the case of first-order kinetics. T,, has also been found to change [5] due to change in no. The latter dependence is an unexpected behaviour and it will lead to give different values of E, and s for various samples of the same system. Also, there is no interconnection between the TL intensities of first- and second-order glow curves represented through (1) and (3). It has been shown [6], however, that in situations when all available electron traps are filled initially, i.e. when N = no, higher-order kinetics (excluding first order) can be represented by a general equation,

where 1 is the order of the kinetics. It is obvious that for 1 = 2 it changes to (3) to represent the TL intensity of the second-order glow curve. Equation (4) fails to represent the TL intensity of the first-order glow curve.

The temperature TM at which the maximum TL intensity in a second-order glow curve appears can be obtained from (3) as

Equations (2) and (5) suggest that they can be represented by

(5)

a general equation [6],

It is apparent from (6) that for 1 = 1, it changes to (2) and for I = 2 it gives the required equation (5).

The Appearance of a Thermoluminescence Glow Curve 283

It is obvious from (1) to (6) that there are some anomalies in the behaviour of TL glow

1. TL intensities of first- and second-order kinetics are not interrelated, 2. TL intensities of second- and higher-order kinetics can be represented by a general

3. surprising dependence of TM on no in second-order kinetics, 4. representation of TM by a general equation, and 5. contradictory assumptions that the electron traps are far from saturation and all

In view of these facts, mechanisms responsible for the appearance of a TL glow curve

curves involving various orders of kinetics, namely:

equation,

electron traps are filled initially.

are reconsidered in this paper with the aim to establish a generalized approach.

2. Proposed Mechanism

The carrier after being thermally released from its trap centre may quickly recombine with an oppositely charged centre. Let us suppose that m and n represent the density of holes and electrons in the recombination centre and trap level, respectively. The TL intensity depends on the extent of recombination which will be reflected through the rate of.decrease of m. Thus, the TL intensity will be given by [3]

where n, is the density of electrons in the conduction band and A , the recombination probability.

The rate of change of n will depend on the excitation of electrons into the conduction band and also on the retrapping of electrons as

dn dt

- - - - ns exp (- $) - ( N - n) n,A, ,

where A, is the retrapping probability. The charge neutrality requirement leads to the condition

m = n + n,. (9)

Equations (7) to (9) are known as Adirovitch set of equations [7] describing the mechanisms responsible for the appearance of a TL glow curve. To have some fruitful information from the Adirovitch set of equations, two basic assumptions have been proposed [3, 71,

dn, dn nc < n and - < ~

dt dt

In view of these assumptions, (9) results in

dm dn m = n and - - -

dt dt -

The aim of introducing the basic assumption n, 4 n, is not to establish a situation in which m becomes equal to n, rather to incorporate the fact that at any instant of time the number

284 J . PRAKASH and D. PRASAD

of electrons in the conduction band is much less than the number of trapped electrons. In view of these arguments it is proposed by Prakash [8] that (9) can be represented as

m = xn + n,, (12)

where x is a dimensionless parameter expressed as 1-1

x = (;) Using (7), (8), (12), and (13), it has been possible to correlate the TL intensities of first- and second-order (1 = 1,2) kinetics. The corresponding general equation, when all the available electron traps are filled initially, is given by

Although the anomalies mentioned in Section 1 are removed through (14), the following points should still be taken into account:

(i) for first-order kinetics x = 1, (ii) irrespective of the order of kinetics involved, one gets x = 1 when n = N , (iii) x depends on n, which itself is a function of time. Thus, x happens to be a variable

quantity. Consequently, x changes during the experimental run of a TL spectrum, (iv) since the order of the kinetics is exclusively expressed by I , it does not seem necessary

to introduce another parameter x for it. If it is done, x and 1 should be related to each other through some suitable constant.

In view of these points it is suggested that the charge neutrality requirement be expressed as

(15) Im = n + n,.

Thus, in the proposed mechanism (7), (8), and (15) constitute a modified Adirovitch set of equations. Substituting the value of 1 in (15), one can analyse TL glow curves of various orders of kinetics.

2.1 First-order kinetics

For first-order kinetics 1 = 1 and hence (15) changes to (9). Incorporating the basic assumptions introduced through (10) and using the modified Adirovitch set of equations, the TL intensity of first-order kinetics can be written as

where suffix 1 represents the parameters associated with the first-order kinetics. Since the first-order kinetics is a recombination dominant process (i.e. mA, + ( N - n) An), one gets from (16)

dn dn I , = - ~ and - = -ns,exp

dt dt

The Appearance of a Thermoluminescence Glow Curve 285

When the system is heated with a linear heating rate b, (17) results in an expression for the TL intensity of the first-order glow curve as

It can be seen that but for suffix 1, (1) and (18) are identical.

2.2 Second-order kinetics

An expression for the TL intensity of second-order kinetics can be derived with the help of the modified Adirovitch set of equations. For second-order kinetics 1 = 2. In view of the two basic assumptions introduced through (lo), one gets from (15)

dm 1 dn 1 2 dt 2 dt (19) - - - ~ m = - n and -

Equations (7), (8), and (19) can be solved for the TL intensity of second-order kinetics as

1 dn 1 2mA, I - - - - = - n s , e x p -- 2 - 2 dt 2 ( :;),,A, + ( N - n)A,’

where suffix 2 represents the parameters associated with the second-order kinetics. In the second-order kinetics, recombination and retrapping processes are equally probable such that mA, = ( N - n) A,. Equation (20) with the help of (19) can hence be expressed as

1 dn dn 2 dt

I - - - - and 2 -

When the system is heated with a linear heating rate b, (21) results in an expression for the TL intensity of the second-order glow curve as

Equation (22) can be compared with (3) for getting an idea about the changes introduced in the intensity of the TL glow curve of second-order kinetics in the proposed mechanism.

2.3 Third-order kinetics

In this case 1 = 3. Equation (15) in view of (10) changes to

1 dm 1 dn

Equations (7), (8), and (23) give the expression for the TL intensity of third-order kinetics as

(24) 1 dn 1 3mA, 3 dt 3

I - - - ~ = -ns3exp - - - ( :;) 3mA, + ( N - n) A n ’ 3 -

286 J. PRAKASH and D. PRASAD

where suffix 3 represents the parameters associated with the third-order kinetics. In the third-order kinetics recombination and retrapping processes are related as 2mA, = ( N - n) A,, and hence (24) with the help of (23) can be expressed as

3 dt dt

when the system is heated with a linear heating rate b, (25) results in an expression for the TL intensity of third-order kinetics as

The TL glow curve involving third-order kinetics is thus expected to be expressed by (26).

2.4 General-order kinetics

Expressions for the intensities of TL glow curves involving higher-order kinetics can similarly be derived following the modified Adirovitch set of equations. In systems involving a general-order kinetics, the processes of recombination and retrapping will be related as (1 - 1) mA, = ( N - n) A,. The TL intensity in such a system will be given by

It is obvious that for I = 1,2, and 3, (27) changes to (18), (22), and (26), respectively. Thus, in the proposed model the TL intensity of higher-order kinetics including first order can be represented with the help of the generalized equation (27).

3. Discussion

In the proposed model, it has been found that the anomalies mentioned in Section 1 in the mechanisms responsible for the appearance of a TL glow curve are removed. A generalized equation has been developed which represents the TL intensity of various-order glow curves. It is obvious from (27) that the location of the TL peak will depend on b similar to the case of first-order kinetics. It is also obvious that the intensity of the TL peak will increase proportionally with the initial concentration no. Further, irrespective of the order of kinetics involved, the location of TM will be independent of the value of no. These results are found to be in agreement with the experimental behaviour recorded in natural barite samples [9] and CaF, : Pr single crystals [lo]. The location of the two peaks observed in the TL spectrum of natural barite samples has been found to be independent of the extent of the irradiation dose. Similarly, the location of the TL peak in CaF, : Pr single crystals is found to be independent of the initial concentration of Pr in CaF,. The intensity of the TL peak in both cases has, however, been found to increase with no as expected.

On the basis of the theory proposed in Section 2, one can tabulate different conditions required for the appearance of a TL glow curve. It is obvious from Table 1 that first-order kinetics is a recombination dominant process. In this process, retrapping is practically

The Appearance of a Thermoluminescence Glow Curve 287

Table 1 Different conditions required for the appearance of a TL glow curve

order of relationship between extcnt of kinetics recombination 1 m and n recombination and (Yo)

retrapping processes

1

2

3

4

5

1

m = 11 m.4, + ( N - n) A , 100 1 2 I 3

m = - n mA, = ( N - n) A , 50

m = - n 2mA, = ( N - n) A,, 33

1 4

m = - n 3mA, = ( N - n) A , 25

1

5 m = - - n 4m.4, = ( N - n) A , 20

negligible in comparison to recombination. This is the process which is frequently observed and widely reported in the literature. In second-order kinetics, recombination and retrapping processes take place with equal probability. It can also be noticed from Table 1 that the concentration of recombination centres is half of the concentration of trapped electrons in second-order kinetics, whereas in first-order kinetics they are expected to be equal. It is also obvious from Table 1 that the extent of recombination for third-and fourth-order kinetics is 33% and 25%, respectively. Thus, the appearance of a TL glow curve can be expressed in terms of the extent of recombination and retrapping processes. It is also obvious that with increasing order of kinetics the extent of recombination decreases with a simultaneous increase in‘ the extent of retrapping.

The location of the peak in a TL glow curve can be ascertained with the help of (17), (21), and (25). These equations can be expressed as

1 dn dn 1 ns, exp (- 2) , I - - - - and - = - - 1 dt dt (21 - 1) 1 -

such that 1 ns, exp (- g) .

I , = ~

(21 - 1) (29)

The location of TL peak can be obtained from (29). It has been found that T,, depends on b, 1, E,, and s, as

(30)

Equations (6) and (30) can be compared for getting an idea about the changes introduced through the proposed model. It is obvious from (30) that for 1 = 1, i.e. for first-order kinetics,

288

80 1

J. PRAKASH and D. PRASAD

Fig. 1. TL glow curves of various orders of kinetics in a hypothetical system with E , = 0.55 eV, s = 2 x lo's-', b = 0.1 Ks-', and no = 2 x 1OI6 m-3. Values of E , and s are assumed to be the same in various orders of kinetics. The number at the curves indicates the involved order of kinetics

it gives (2). For second-order kinetics, i.e. for 1 = 2, (30) results in

It is apparent from (31) that TM2 is independent of no as expected. It is also obvious from (2) and (31) that TMl and TM2 will not be located at the same position. In a hypothetical system with given values of b, E,, s, and no, the location of the TL peak has been found to change due to a change in the order of the kinetics as shown in Fig. 1. It is obvious from the figure that the TL spectrum gets broadened with the increase in the order of kinetics. The location of the TL peak has been found to shift towards higher temperature along with a simultaneous decrease in the intensity of the TL peak due to an increase in the value of 1. It is also obvious that the extent of these changes is more pronounced when the order of the kinetics changes from 1 = 1 to 2. An idea about the changes in TM and I , can be gained from Fig. 2. Thus, depending on the order of kinetics involved in the system under investigation, the TL spectrum should be in accordance with the curves shown in Fig. 1.

The two glow curves corresponding to first- and second-order kinetics may appear at the same location. This possibility is purely hypothetical and will seldom arise in practice. However, for such as possibility conditions can be established from (2) and (31). When TL data are recorded with the same heating rate in both cases, one gets from (2) and (31) the required conditions for T M , = TM2 = TM as

In the cases when E,, = Ea2, (32) gives

2s, = 3s,. (33)

The Appearance of a Thermoluminescence Glow Curve 289

260 I I I I

- - I

rn I

v)

- 4 0 5 7

9 - z

H

0

Fig. 2. Dependence of T, and I , on the order of kinetics for the hypothetical system of Fig. 1

Thus, the peaks of the TL glow curves of first- and second-order kinetics recorded with the same heating rate will appear at the same position when either E,, = E,, and 3s, = 2s, or their E , and s values are related through (32).

The solution of (28) is found to be

1, = ~~~ 1 nosI exp I t s , exp (- 2)). (21 - 1)

(34)

It is obvious, therefore, that irrespective of the order of the kinetics involved, the TL intensity decays exponentially. The decay is fastest in first-order kinetics as shown in Fig. 3. The extent of exponential decay has been found to decrease with the increase in the order of kinetics. In the proposed mechanism, second-order kinetics is found to be a more slowly decaying process than the first-order kinetics, and the third-order kinetics decays more slowly than the second-order kinetics, and so on.

Fig. 3. Isothermal decay in a hypothet- ical system involving various order of kinetics. Values of E , and s are assumed to be the same in various orders of kinetics. E , = 0.55 eV, s = 2 x 10*sC1, no = 2 x 10l6 m-3, and T = 300 K. The number at the curves indicates the in- volved order of kinetics

19 physica (a) 142/1

290 J. PRAKASH and D. PRASAD: The Appearance of a Thermoluminescence Glow Curve

4. Conclusions

A theoretical model has been developed which explains the mechanisms responsible for the appearance of a TL glow curve involving various orders of kinetics. The Adirovitch set of equations has been modified. A generalized equation has been developed which is found to be capable of explaining the TL intensity of various orders of kinetics including the first order. The anomalies mentioned in Section 1 have been removed through the suggested mechanisms. It has been found that irrespective of the order of the kinetics involved, the TL intensity decays exponentially. The decay is found to be fastest for the first-order kinetics. It has also been found that the first-order kinetics is a recombination dominant process with negligible retrapping. The extent of recombination and retrapping processes decides on the order of the kinetics involved. With increasing order of kinetics the extent of recombination decreases with a simultaneous increase in the extent of retrapping.

Acknowledgements

The authors are thankful to Dr. A. K. Nishad for his interest in the work. Valuable suggestions received through Prof. R. S. Meltzer are also thankfully acknowledged.

References

11 J. T. RANDALL and M. H. F. WILKINS, Proc. Roy. Soc. A 184, 366 (1945). 21 G. F. J. GARLICK and A. F. GIBSON, Proc. Roy. SOC. 60, 574 (1948). 31 R. CHEN and Y. KIRSH, Analysis ofThermally Stimulated Processes, Pergamon Press, Oxford 1981. 41 W. HOOGENSTRAATEN, Philips Res. Rep. 13, 515 (1958). 51 R. CFIEN, D. J. HUNTLEY, and G. W. BERGER, phys. stat. sol. (a) 79, 251 (1983). .6] R. CHEN and S. A. A. WINER, J . appl. Phys. 41, 5227 (1970). 71 E. I. ADIROVITCH, J. Phys. Rad. 17, 705 (1956). 81 J. PRAKASH, Solid State Commun. 85, 647 (1993). 91 M . P R O K I ~ , J . Phys. Chem. Solids 18, 617 (1977). 01 R. K. SINHA and M. I . MUKHERJEE, phys. stat. sol. (b) 105, 69 (1981).

(Received Februury 15, 1993; in revisedjorrn August 16, 1993)