GENERAL 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.

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.

We have already seen in this web site the two special cases of the OTOR model which lead to 1st and 2nd order Thermoluminescence curves. The differential equations for those two cases are:

The above equations have be generalized for a situation that would be "intermediate" between 1st and 2nd order kinetics.
This is known as the GENERAL ORDER kinetics, and is described by the following differential equation:

The coefficient b here denotes the "general order" of the kinetics, and has values between 1 and 2.
Notice that for b=1 the general order equation reduces to 1st order kinetics, and for b=2 it reduces to 2nd 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, but they are also being retrapped in the electron trap and they are also recombining 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.

By assuming a linear rate of heating of the sample, the above differential equation can be solved to yield the following solution:

A graph of this equation is shown below.

THE EFFECT OF THE GENERAL ORDER COEFFICENT b ON THE TL GLOW CURVES

Here is an example of 3 general order TL curves, calculated using 3 different values of the general order coefficient b.

It must be noted that the general order equation for TL glow curves is a semi-empirical equation, based on a generalization of the first and second order kinetics equations (b=1,2).

In the links below you can find:

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

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

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