First order kinetics theory

 

FIRST ORDER KINETICS model

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 FIRST 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 negligible compared to the probability of recombination.

By using these approximations, we can arrive at the following simple differential equation for first order kinetics . This equation describes the rate of change of the concentration of trapped electrons n(t) during the thermoluminescence measurement:

 

 

 

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, but the probability for retrapping is assumed to be much smaller than the probability of recombination at 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+βt, where To=room temperature, and β=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, and k=Boltzmann constant.

A graph of this equation is shown below.

THE ASYMMETRY OF FIRST ORDER TL GLOW CURVES

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

 

 

 

 

 

 

Notice that this curve (known as a TL GLOW CURVE), is not symmetric. Its asymmetry can be expressed by measuring the temperatures Tmax, T1 and T2 shown in the figure above. Here Tmax=temperature of maximum TL intensity and T1,T2=temperatures at half the maximum TL intensity.

It is customary to define the quantities τ=Tmax-T1 , δ=T2-Tm, and ω=T2-T1.

The asymmetry factor μ for this FIRST ORDER TL GLOW CURVE is ALWAYS equal to μ=δ/ω=0.42

THE EFFECT OF no ON FIRST ORDER TL GLOW CURVES

Here is an example of 3 first 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 only the maximum height of the TL glow curve, and leaves the shape of the 1st order TL glow curve and the temperature of maximum TL intensity (Tmax) unchanged.

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

Here is an example of 3 first order TL curves, calculated using 3 different values of the activation energy E=0.90, 0.95 and 1.0 eV. Notice that as the energy E is increased, the TL glow curve shifts towards higher temperatures, but the curve maintains its overall shape.

 

 

 

 

Here is a LINK TO many Mathematica programs

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