SECOND ORDER KINETICS MODEL IN THERMOLUMINESCENCE

The simplest model in Thermoluminescence consists of two energy levels: the electron traps and the recombination center (RC) shown in the figure below. This is also known as the One-Trap-One-Recombination-Center (OTOR) model.

A special case of the OTOR model leads to Thermoluminescence curves described by what is known as SECOND ORDER KINETICS, is discussed below.

LIST OF VARIABLES USED IN THE OTOR MODEL

N=total concentration of the electron traps in the crystal (in cm^-3).
n=concentration of the filled electron traps in the crystal (in cm^-3).
nc=concentration of the free carriers in the conduction band CB (in cm^-3).
E=activation energy of the electron traps (in eV).
s=frequency factor of the electron trap (in s^-1).
An=capture coefficient of the traps (in cm^3. s^-1).
Ah=capture coefficient of the recombination center RC (in cm^3. s^-1).

For more details on the OTOR model see, for example, the book : Chen, R. and McKeever, S.W.S. 1997. Theory of thermoluminescence and related phenomena. World Scientific, Singapore, Chapter 4.

It is posssible to arrive at an exact solution to the OTOR model, by imposing the following simplifying approximations:
(a) The QUASISTATIC EQUILIBRIUM conditions: nc < < n and dnc/dt < < dn/dt
(b) The probability of retrapping is the dominant process, being much larger than the probability of recombination.
(c) The two capture coefficients in the RC and in the trap are equal: An=Ah

By using these approximations, we can arrive at the following simple differential equation for second order kinetics .

THE PHYSICS BEHIND THE DIFFERENTIAL EQUATION

Here dn/dt represent the rate of change of concentration of electrons n(t) as the sample is heated during the thermoluminescence measurement.
The electrons leave the traps via thermal excitation, which is described mathematically by the term [n.s.exp(-E/kT)]
The electrons are also being retrapped in the trap, and the probability for retrapping is assumed to be very large, so that the dominating process in 2nd order kinetics is retrapping, instead of recombination in the RC.

The observed TL intensity will be equal to the negative rate of change of the concentration of electrons in the trap: TL=-dn/dt.

During a typical TL measurement, the temperature T is changed linearly with the time t, so that T(t)=To+bt, where To=room temperature, and b=heating rate in C/s .
By assuming a linear rate of heating of the sample b, the above differential equation can be solved to yield the following solution:

Here no=initial concentration of filled traps at time t=0 (in cm^-3), and k=Boltzmann constant.

A graph of this equation is shown below.

THE ASYMMETRY OF SECOND ORDER TL GLOW CURVES

Here is an example of a second order TL curve, calculated by solving the above differential equation with the following parameters: E=1 eV, s=10^12 s^-1, no=N=10^10 cm^-3.

Notice that this SECOND ORDER TL GLOW CURVE, is much more symmetric than the first order kinetics peak. Just like in the case of the first order TL peak, we can measure the temperatures Tmax=temperature of maximum TL intensity and T1,T2=temperatures at half the maximum TL intensity.

Again we can define the quantities t=Tmax-T1 , d=T2-Tm, and w=T2-T1.

The asymmetry factor for a SECOND ORDER TL GLOW CURVE is always equal to
m=d/w=0.52

THE EFFECT OF no ON SECOND ORDER TL GLOW CURVES

Here is an example of 3 second order TL curves, calculated for several values of the initial concentration of filled traps no.

The solution is obtained with the following parameters: E=1 eV, s=10^12 s^-1, N=10^10 cm^-3 and no/N=1.0, 0.5, 0.1 .

We notice that 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.

COMPARISON OF 1ST AND 2ND ORDER TL GLOW CURVES

Here is a graph comparing the shapes of 1st and 2nd order TL curves. Both curves have been normalized by dividing each TL glow curve by its maximum TL value. (red curve=1st order, green=2nd order)

THE EFFECT OF THE ACTIVATION ENERGY E ON THE TL GLOW CURVES

Here is an example of 3 second order TL curves, calculated using 3 different values of the activation energy E.

In the links below you can find:

(a) The listing of a Mathematica program that solves numerically the differential equation and produces the second order thermoluminescence graphs shown above

(b) A well-documented version of the same Mathematica program

	Mathematica program for Second Order Kinetics model
	Detailed documentation of the Second Order Kinetics program
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