The OTOR model in TL

The OTOR 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.

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.

and also, for example, in the following paper :
Sunta, C.M., Feria, Ayta W.E., Piters, T.M., Watanabe, S., 1999. Limitation of peak fitting and peak shape methods for determination of activation energy of thermoluminescence glow peaks. Radiation Measurements 30, 197 – 201.

The differential equations governing the traffic of electrons between the trap level, the recombination center and the conduction band are:

THE PHYSICS BEHIND THE DIFFERENTIAL EQUATIONS

 

The first equation describes the traffic of electrons in and out of the electron trap.
The electrons can leave the traps via thermal excitation, which is described mathematically by the term [n.s.exp(-E/kT)]
The electrons can also be retrapped in the trap, an event described mathematically by the retrapping term [nc.(N-n).An].

The second equation describes the traffic of electrons in and out of the conduction band.
The electrons in the conduction band can be trapped in the recombination center RC, an event described mathematically by the term [nc.(n+nc).Ah]
The quantity (n+nc) here represents the total concentration of FILLED TRAPS in the system at any moment. Because of conservation of charge, this quantity (n+nc) is also equal to the total concentration of FILLED HOLES in the recombination center.

The third equation above gives the observed TL, which is proportional to the amount of light measured during the thermoluminescence measurement.

Unfortunately these differential equations can not be solved in any closed form, and the solutions must be obtained numerically.
We will now present two examples of the OTOR solutions, for different values of the parameters.

EXAMPLE #1: OTOR

The parameters for the calculation shown below are:
An/Ah=0.01 and no/N=1, N=10^10, An=10^-7, heating rate=1 C.s^-1

In this example the coefficient of retrapping An is 100 times smaller than the coefficient of recombination (An/Ah=0.01), and the traps are initially full (no/N=1). The result of the OTOR calculation produces a TL glow curve which has the shape of a 1st order kinetics curve.

In this example τ=Tmax-T1=17.3 C , δ=T2-Tm=12.5 C, and ω=T2-T1=29.9 C.

The asymmetry factor μ for this OTOR TL glow curve is equal to μ=δ/ω=12.5/29.9=0.42, which corresponds to the shape of a 1st order glow curve.

EXAMPLE #2: OTOR

 

The parameters for the calculation shown below are:
An/Ah=1 and no/N=1, N=10^10, An=10^-7, heating rate=1 C.s^-1

In this example the coefficient of retrapping An is equal to the coefficient of recombination (An/Ah=1), and the traps are initially full (no/N=1). The result of the OTOR calculation produces a TL glow curve which has a shape close to the shape of a 2nd order kinetics curve.

In this example τ=Tmax-T1=20.0 C , δ=T2-Tm=20.3 C, and ω=T2-T1=42.1 C.

The asymmetry factor μ for this OTOR TL glow curve is equal to μ=δ/ω=12.5/29.9=0.48, which is close to the shape of a 2nd order glow curve (m=d/w=0.52).

The following is a Mathematica program to solve the system of differential equations for the OTOR model.

 

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