Theory of Thermoluminescence
by Dr. Reuven Chen
Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel.
The basic theory of thermoluminesence (TL) is based on the occurrence of imperfections, impurities, and defects, found within an insulating material. These lattice sites may capture electrons and holes during the excitation of the sample and later, during the heating, these charge carriers can recombine and produce the emission of light in the form of a TL glow curve. The process leading to recombination includes, in many cases, the transition of charge carriers through the conduction or valence band, but localized transitions may also take place. In most cases, the theory consists of solving the relevant sets of coupled differential equations,either by using some simplifying assumptions or by solving numerically the equations for certain sets of trapping parameters.
1 Introduction
The effect of thermoluminescence (TL) is the emission of light from solids, usually crystalline insulators, following excitation, usually by some irradiation. Energy is absorbed in the sample during the excitation, and released during the heating, yielding a glow curve, namely, a graph of emitted-light intensity versus temperature. The glow curve usually includes one or more glow peaks that may be either separate or overlapping. The emitted light may include different spectral components which indicate different transitions taking place during the heating. Practically always, the glow curve can be detected only following a first heating, and a subsequent heating does not produce any light emission until another irradiation takes place. The main application of TL, along with the closely related effect of optically stimulated luminescence (OSL) is in dosimetry. The dependence of the measured luminescence on the preceding excitation dose should be taken into consideration; linear dose dependence is desirable, but other dependencies often occur. Different dose-dependence behaviours are observed when using different sources of radiation. [math]\alpha[/math], [math] \beta [/math], and [math]\gamma[/math] irradiations as well as X-rays, UV light, high-energy particles and neutron beams may be used to induce TL in different crystals. Another application of TL, derived from the property of dose dependence is the dating of archaeological and geological samples. An important subject in the study of TL has to do with the evaluation of the trapping parameters, mainly the activation energy and the frequency factor of the involved traps. Also the stability of the TL signal at ambient temperature has been studied extensively. Normal thermal fading as well as anomalous fading have been observed in different materials and a number of theoretical explanations have been provided. TL has been studied in hundreds of different materials in the quest for an optimal material for TL dosimetry.
2 Basic Theory
The main requirement for producing a glow peak is the occurrence of two imperfections in the lattice. During excitation, one is capable of trapping an electron and the other traps a hole. A schematic energy level diagram is shown in Fig. 1. The basic theory of TL was first introduced by Randall and Wilkins (1945). These authors assumed that during excitation by irradiation, electrons are trapped in an electron trap of concentration N (cm[math]^{-3}[/math]), and holes are trapped in hole centres, M (cm[math]^{-3}[/math]). During heating, electrons are raised thermally into the conduction band and, according to this basic theory, they recombine almost immediately with a hole in a centre to produce photons. The basic differential equation for this process was given by Randall and Wilkins as a first-order equation
[math]I(T)=-\frac{\mathrm{dn} }{\mathrm{d} t}=sn\, exp\left ( -E/kT \right )\tag{1}[/math]
where n (cm[math]^{-3}[/math]) is the instantaneous concentration of trapped electrons, E (eV) the activation energy for releasing trapped electrons, s ([math]s^{-1}[/math]) the frequency factor, k (eV/K) Boltzmann's constant, T (K) the temperature, t (s) is the time and I the emitted intensity. As is, the units of I(T) are cm[math]^{-3}s^{-1}[/math], however, a dimensional constant should be added which has been set arbitrarily to unity. With this constant, I(T) will be in units of photons per second or energy per second. In order to solve this equation, one should use a heating function that relates temperature and time. In many cases, a linear function is used, [math]T=T_{0}+\beta t[/math] where [math]\beta[/math] (K/s) is the constant heating rate and [math]T_{0}[/math] the initial temperature. The solution of Eq. (1) is [math]I(T)=n_{0}s\, exp\left ( -E/kT \right )\, exp\left [ -\frac{s}{\beta }\int_{T_{0}}^{T}exp\left ( -E/k\theta \right )d\theta \right ]\tag{2}[/math] where [math]n_{0}[/math] is the concentration of trapped electrons at the beginning of heating. This function represents an asymmetric peak with a slower increase and faster decrease. Figure 2 depicts schematically the TL intensity in a first-order TL peak. Here, [math]T_{m}[/math] is the maximum temperature and [math]T_{1}[/math] and [math]T_{2}[/math] are the low- and high-temperature half intensity temperatures, respectively. [math]\alpha[/math] and <math\beta</math>are the low- and high-temperature half peaks and [math]\omega[/math] is the full half peak. The symmetry of the peak is usually measured by the symmetry factor [math]\mu _{g}=\frac{\delta }{\omega }[/math] which for the first-order case is typically ~0.42 (see e.g. Chen and McKeever, 1997). The condition for the maximum of a first-order peak is reached by setting the derivative of Eq. (2) to zero which yields [math]\frac{\beta E}{kT_{m}^{2}}=s\, exp\left ( -E/kT_{m} \right )\tag{3}[/math] In a more comprehensive approach, one should consider the possibility of retrapping of freed electrons from the conduction band into the original trap which has been described by Halperin and Braner (1960). The processes taking place during the heating of the sample, when one trap and one centre are involved, are shown in Fig. 1. The governing equations are
[math]I=-\frac{dm}{dt}=A_{m}m_n{c}\tag{4}[/math]
[math]-\frac{dn}{dt}=sn\, exp\left ( -E/kT \right )-n_{c}(N-n)A_{n}\tag{5}[/math]
[math]\frac{dn_{c}}{dt}=sn\, exp\left ( -E/kT \right )-n_{c}\left [ (mA_{m}+(N-n)A_{n} \right ]\tag{6}[/math]
where [math]A_{m}[/math] (cm[math]^{3}[/math]s[math]^{-1}[/math]) and A[math]_{n}[/math] (cm[math]^{3}[/math]s[math]^{-1}[/math]) are respectively the recombination- and retrapping-probability coefficients. Halperin and Braner (1960) made the well-known "quasi-equilibrium" assumptions, namely
[math]\left | \frac{dn_{c}}{dt} \right |\ll \left | \frac{dn}{dt} \right |,\left | \frac{dm}{dt} \right |;\; n_{c}\ll n,m\tag{7}[/math]
and reached the simplified equation
[math]I=-\frac{dm}{dt}=sn\, exp\left ( -E/kT \right )\frac{A_{m}m}{A_{m}m+A_{n}(N-n)}\tag{8}[/math]
In the simple case where there is only one trap and one centre, n=m and Eq. (8) reduces to
[math]I=-\frac{dn}{dt}=sn\, exp\left ( -E/kT \right )\frac{A_{m}n}{A_{m}n+A_{n}(N-n)}\tag{9}[/math]
which is an equation with one unknown function, [math]n(t)[/math]. Further possibilities of simplifying Eq. (9) have to do with different relations between recombination and retrapping. If recombination dominates, [math]A_{m}m\gg A_{n}(N-n)[/math], then Eq. (9) reduces to the first-order case of Eq. (1). If, on the other hand, retrapping dominates [math]A_{m}m\ll A_ {n}(N-n)[/math], Eq. (9) reduces to
[math]I=-\frac{dn}{dt}=\frac{sA_{m}}{NA_{n}}n^{2}\, exp\left ( -E/kT \right )\tag{10}[/math]
which is a second-order TL equation. Alternatively, if one assumes (see e.g. Garlick and Gibson, 1948), equal probabilities for recombination and retrapping, [math]A_{m}=A_{n}[/math], one gets from Eq. (9)
[math]I=-\frac{dn}{dt}=\frac{s}{N}n^{2}\, exp\left ( -E/kT \right )\tag{11}[/math]
which is also a second-order equation. Since the trap and centre are two independent entities, the latter condition for second-order kinetics is less likely to occur than the former. Both equations (10) and (12) can be written in a combined way
[math]I=-\frac{dn}{dt}=s'exp\left ( -E/kT \right )n^{2}\tag{12}[/math]
where [math]s'[/math] is a "pre-exponential" factor with units of cm[math]^{3}s^{-1}[/math]. The solution of Eqs. (10-12) is a peak-shaped curve which is nearly symmetric, with a typical shape factor [math]\mu _{g}\sim 0.52[/math]. Glow peaks with intermediate symmetries between that of first- and second-order have been observed, prompting May and Partridge (1964) to suggest the use of "general-order" kinetics, namely, using the equation
[math]I=-\frac{dn}{dt}=s'exp\left ( -E/kT \right )n^{b}\tag{13}[/math]
where [math]b[/math] is the order of the kinetics which may be between one and two. The solution of this equation is a peak-shaped curve with a symmetry factor between 0.42 and 0.52. It should be noted however that Eq. (13) is merely heuristic in the sense that it cannot be reached from the initial equations (4-6). Moreover, the units of the pre-exponential factor [math]s'[/math] in the general-order case are cm[math]^{3(b-1)}s^{-1}[/math]. In the first- and second-order cases, this yields the known units of these factors but it does not have much physical meaning when [math]b\neq 1[/math] and [math]b\neq 2[/math]. An alternative way of presenting TL peaks with kinetics intermediate between first- and second-order has been suggested by Chen et al. (1981), namely, the "mixed-order kinetics" given by
[math]I=-\frac{dn}{dt}=s'n\left ( n+c \right )exp\left ( -E/kT \right )\tag{14}[/math]
where [math]c[/math] is a constant associated with the concentration of disconnected traps or centres and [math]s'[/math] is a pre-exponential factor with units of cm[math]^{3}[/math]s[math]^{-1}[/math] like in the second-order case. Obviously, [math]c\gg n[/math] results in the first-order equation and [math]c\ll n[/math] leads to the second-order case. Of course, different intermediate cases are possible.
3 Methods for Evaluating Trapping Parameters
A number of methods have been developed for evaluating the activation energy from the shape of the TL peak. The simplest one appears to be the "initial-rise" method. If we consider, for example, Eq. (1), at the low temperature end of the peak, n varies only slightly, and we can write
[math]I(T)\propto exp\left ( -E/kT \right )\tag{15}[/math]
Garlick and Gibson (1948) suggested to plot [math]\ln (I)[/math] as a function of [math]1/T[/math] in the low temperature range which should yield a straight line, the slope of which is [math]–E/k[/math]; from this, [math]E[/math] can be calculated. The method is considered to be generally applicable although some "pathological" examples have been reported in the literature. In all the instances described in Eqs. (8-14), as long as [math]n[/math] and [math]m[/math] vary only slightly in the initial-rise range, the described method is expected to yield an acceptable value for E. The main problem here is that the initial-rise range is limited to, say, 5% of the maximum intensity and in low-intensity peaks this may be a significant restriction. Several methods have been developed based on various shape features of the TL peak. As an example, for a first-order peak one can use the formula
[math]E=2.52\frac{kT_{2}^{m}}{\omega }+2kT_{m}{tag{16}[/math]
where [math]\omega[/math] is the full width defined above. Extension of this formula to non-first-order peaks use interpolation on the symmetry factor [math]\mu_{g}[/math] (see e.g. P. 66 in Chen and Pagonis, 2011).
