The IMTS model in THERMOLUMINESCENCE
A more complex model in Thermoluminescence consists of three energy levels: the electron traps, the recombination center (RC), and a thermally-disconnected-deeper-trap (TDDT), shown in the figure below.
LIST OF VARIABLES USED IN THE IMTS 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).
M=total concentration of the electron traps in the TDDT’s (in cm^-3).
m=concentration of the filled electron traps in the TDDT’s (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).
Am=capture coefficient of the TDDT’s (in cm^3. s^-1).
For more details on the IMTS model see, for example, the book :
Chen, R. and McKeever, S.W.S. 1997. Theory of thermoluminescence and related phenomena. World Scientific, Singapore.
For a comparison of the OTOR and IMTS models see, for example, 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.
Also notice that the OTOR model is a special case of the IMTS model described here, with Am=0, M=0 and m=0.
The differential equations governing the traffic of electrons between the trap level, the recombination center, the conduction band and the TDDT are:
THE PHYSICS BEHIND THE DIFFERENTIAL EQUATIONS
The first equation describes the traffic of electrons in and out of the electron trap, just as in the simpler OTOR model. 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 from the conduction band into the TDDT. The trapping of the electrons in the TDDT is described mathematically by the term [nc.(M-m).Am].
The third 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.(m+n+nc).Ah]
The quantity (m+n+nc) here represents the total concentration of FILLED TRAPS in the system at any moment. Because of conservation of charge, this quantity (m+n+nc) is also equal to the total concentration of FILLED HOLES in the recombination center.
The last 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 several examples of the IMTS solutions, for various values of the parameters.
Below you can find the listing of a Mathematica program that solves the IMTS system of differential equations