**The Zimmerman model for Thermoluminescence of Quartz**

**Approximately 40 years ago, J.Zimmerman suggested a model which explains several observed thermoluminescence properties of quartz. The original Zimmerman model was modified by R. Chen by adding an extra energy level. For more details on the Zimmerman model and its importance in the Predose dating technique, see the References at the end of this web page. **

**The Zimmerman model consists of four energy levels: the main electron trap T, a deeper thermally-disconnected electron trap S, the recombination center L, and a hole reservoir R.
These four levels together with the conduction band, the valence band, and the possible transitions in the model are shown in the figure below. **

**In this web page you will find a Mathematica program which simulates 3 stages which are typical of a TL measurement. In the first stage the quartz sample is irradiated with ionizing radiation (e.g. beta radiation); this is the “irradiation stage”. In the second stage the sample is allowed to remain (“relax”) at room temperature for a minute or so; this is the “relaxation stage”. Finally in the third stage the sample is heated in order to measure the TL curve; this is the “heating stage”. **

** **

**LISTING OF THE PARAMETERS IN THE ZIMMERMAN MODEL**

**A. CARRIER CONCENTRATIONS **

**Nt= total electron concentration of the main trapping center T (in cm^-3).
Ns= total electron concentration of the competing trapping center S (in cm^-3).
Nr=total hole concentration on the hole reservoir R (in cm^-3)
M=total hole concentration of the luminescence center L (in cm^-3)
Also nt , ns, nr and m are the corresponding instantaneous occupancies of the 4 levels in the model (in cm^-3)
Finally nc, nv (in cm^-3) represent the concentrations of electrons and holes in the conduction and valence band, respectively. **

**B. TRANSITION PROBABILITIES, ENERGIES, FREQUENCY FACTORS **

**The activation energy for the main traps T is Et (in eV) and its frequency factor is st ( in s^-1)
The competitor traps S are considered to be thermally disconnected; electrons can be trapped but can not escape thermally from S.
The activation energy for the hole reservoir R is Er (in eV) and its frequency factor is sr (in s^-1).
The retrapping probability coefficients for R, T and S are denoted by Ar, At and As (in cm^3. s^-1)
Am, A1 (cm^3 s-1) are, respectively, the recombination probability coefficients of electrons and holes into the recombination center L. **

**C. IRRADIATION PARAMETERS **

**The rate of production of electron-hole pairs x (in cm^-3. s^-1) is proportional to the dose rate
The dose is given by D=x.t, where t is the irradiation time in sec and D is in cm^-3. **

**More details on the model can be found in the following paper :
Chen, R. and Leung, P.L., 1999. Modeling the pre-dose effect in thermoluminescence.
Radiation Protection Dosimetry, 84, 43-46. **

** THE DIFFERENTIAL EQUATIONS-IRRADIATION STAGE**

**The differential equations governing the traffic of electrons between the trap level T, the competitor S, the recombination center L, the resrvoir R, the conduction band and the valence band during the IRRADIATION STAGE of a quartz sample are: **

** **

** THE PHYSICS BEHIND THE DIFFERENTIAL EQUATIONS**

**The first equation describes the traffic of electrons in and out of the electron trap T. The electrons can leave the traps via thermal excitation, which is described mathematically by the term [nt.st.exp(-Et/kT)]. Here we are assuming that irradiation is taking place at room temperature, so that this term is essentially zero and is omitted. The electrons can also be retrapped in T, an event described mathematically by the retrapping term [nc.(Nt-nt).At]. **

**The second equation describes the traffic of electrons into the competitor trap S. The CB electrons can be trapped into S, an event described by the term [nc.(Ns-ns).As]. Since this trap is considered thermaly disconnected, the electrons can not be thermally released from S. **

**The third equation describes the traffic of electrons and holes into the recombination (luminescence) center L. The electrons can be trapped into L, an event described by the term -[nc.m. Am]. The minus tells us that the capturing of the electrons from the CB leads to a reduction in the concentration of holes in L (i.e. dm/dt will be negative due to the capturing of electrons).
The other event taking place at L is the capturing of holes from the valence band, described by the term [+A1. nv. (M-m)]. The plus tells us that the capturing of the holes from the VB leads to a increase of the concentration of holes in L (i.e. dm/dt will be positive due to the capturing of holes). **

**The fourth equation describes the rate of change of the concentration of holes in the VB, dnv/dt. Some of the holes at the VB are captured in the luminescence center, an event described by the term [-A1. nv. (M-m)]. Notice that this term is now negative, since it leads to a reduction of the number of holes (nv) in the VB.
Some of the holes in the VB can also be captured in the hole reservoir R; this capturing is decribed by the term [-Ar. nv. (Nr-nr)]. Again this term is negative, since it leads to a reduction of the concentration of holes (nv) in the VB.
Finally, this equation contains the constant term R, which describes the creation of holes in the VB by the ionizing radiation. This term is positive, since it leads to an increase of the concentration of holes (nv) in the VB. **

**The fifth equation describes the rate of change of the concentration of electrons in the CB, dnc/dt. Some of the electrons at the CB are captured in the luminescence center, an event described by the term [-Am. m. nc]. Notice that this term is now negative, since it leads to a reduction of the concentration of electrons (nc) in the CB.
Some of the electrons in the CB can also be captured in the competitor trap S; this capturing is decribed by the term [-As. nc. (Ns-ns)]. Again this term is negative, since it leads to a reduction of nc.
It is asumed that at room temperature, none of the electrons trapped in T are thermally released in the CB.
Finally, this equation contains the constant term R, which describes the creation of a new number of electrons in the CB by the ionizing radiation. This term is positive, since it leads to an increase of the concentration of electrons (nc) in the CB. **

**The sixth equation describeds the rate of change of the concentration of holes in the reservoir R (dnr/dt). The holes can leave the reservoir R via thermal excitation, which is described mathematically by the term [nr.sr.exp(-Er/kT)]. Again we are assuming irradiation taking place at room temeprature, so that this term is essentially zero and is omitted during the irradiation stage.
The holes can also be retrapped in R from the VB, an event described mathematically by the retrapping term [Ar.(Nr-nr).nv]. **

**This system of 6 equations must be solved from time t=0 to time t=irradiation time of the sample. **

**The figure below shows the concentrations of nc,nv,m,nr,nt,ns as a function of the time t during the irradiation stage. **

** **

**The final concentrations of nc,nv,nr,ns,nt,m at the end of the irradiation stage, are used as initial concentrations for the next stage, the relaxation stage. **

** THE DIFFERENTIAL EQUATIONS-RELAXATION STAGE**

**The differential equations governing the traffic of electrons between the trap level T, the competitor S, the recombination center L, the resrvoir R, the conduction band and the valence band during the RELAXATION STAGE of a quartz sample are: **

** **

** THE PHYSICS BEHIND THE DIFFERENTIAL EQUATIONS**

**Notice that these are exactly the same system of equations as in the irradiation stage described above, but the ionizing irradiation term R has been set equal to R=0. **

**This system of 6 equations must be solved from time t=0 to time t=relaxation time of the sample at room temperature. In most situations the concentrations of nr,ns,nt,m will change very little during the relaxation stage, while the concentration of free carriers in the CB and VB will go to zero at the end of the relaxation period. **

**A time of t=60 seconds turns out to be large enough for the concentrations of electrons and holes in the CB and VB correspondingly to go to zero. **

**The final concentrations of nc,nv,nr,ns,nt,m at the end of the relaxation stage, are used as initial concentrations for the next stage, the heating stage. **

**The figure below shows the concentrations of nc,nv,m,nr,nt,ns as a function of the time t during the relaxation stage. **

** **

** THE DIFFERENTIAL EQUATIONS-HEATING STAGE **

**The differential equations governing the traffic of electrons between the trap level T, the competitor S, the recombination center L, the resrvoir R, the conduction band and the valence band during the HEATING STAGE of a quartz sample are: **

** **

** THE PHYSICS BEHIND THE DIFFERENTIAL EQUATIONS**

**Notice that during the heating stage it is more convenient to integrate the quantities dn/dT instead of dn/dt, i.e. it is much easier to use temperature istead of time as the variable of integration. This change of variable results in all the terms in the differential equations now being divided by the heating rate b=dT/dt. **

**The first equation describes the traffic of electrons in and out of the electron trap T as before, but now the electrons can also leave the traps via thermal excitation during the heating stage. The thermal excitation is described mathematically by the new term [nt.st.exp(-Et/kT)]. The electrons can also be retrapped in T as before, an event described mathematically by the retrapping term [nc.(Nt-nt).At]. **

**The second equation describes the traffic of electrons into the competitor trap S and remains essentially unchanged. **

**The third equation describes the traffic of electrons and holes into the recombination (luminescence) center L as before. The electrons can be trapped into L (the term -[nc.m. Am]), and the holes are captured from the valence band, as described by the term [+A1. nv. (M-m)]. **

**The fourth equation describes the traffic of holes in the VB, dnv/dt. Some of the holes at the VB are captured in the luminescence center (the term [-A1. nv. (M-m)]), while some of these holes are captured in the hole reservoir R; this capturing is decribed by the term [-Ar. nv. (Nr-nr)]. **

**The fifth equation describes the rate of change of the concentration of electrons in the CB, dnc/dt, as before. Some of the electrons at the CB are captured in the luminescence center, (the term [-Am. m. nc]), while some of the electrons in the CB can also be captured in the competitor trap S; this capturing is decribed by the term [-As. nc. (Ns-ns)]. **

**The sixth equation describeds the rate of change of the concentration of holes in the reservoir R (dnr/dt). During the heating stage the holes can leave the reservoir R via thermal excitation, which is described mathematically by the new term [nr.sr.exp(-Er/kT)].
The holes can also be retrapped in R from the VB, an event described mathematically by the retrapping term [Ar.(Nr-nr).nv]. **

**This system of 6 equations must be solved from q temperature T=room temperature=20 C, to a temperature=200 C (for example). **

**The figure below shows the concentrations of nc,nv,m,nr,nt,ns as a function of the temperature T during the heating stage. **

** **

**And finally here is the TL glow curve measured during the heating stage. **

** **

**Below you can find the listing of a Mathematica program that solves the above system of differential equations:**

Converted by *Mathematica* February 23, 2003