Читать книгу A Course in Luminescence Measurements and Analyses for Radiation Dosimetry - Stephen W. S. McKeever - Страница 35

If, instead of heating a material, the trapped electrons are released from their traps via absorption of energy from photons, Equation 2.1 now becomes:

(2.12)

where Φ is the intensity of the stimulating light (in units of m^–2s^–1) and σp(E) is the photoionization cross-section (m²) for a stimulation energy E. If Eo is the threshold photon energy required to excite the electron from the trap (i.e. the optical trap depth) one might expect Eo = Et, that is, the thermal trap depth Et and the optical trap depth Eo are the same. However, thermal energy is also absorbed by lattice phonons such that:

(2.13)

where Eph is the phonon energy given by:

(2.14)

Here, S is the Huang-Rhys factor, h is Planck’s constant and vph is the phonon vibration frequency.

The usual way to consider the differences between the optical trap depth Eo and the thermal trap depth Et, is to use the configurational coordinate diagram, shown in Figure 2.8, where potential energy curves for the electron in the trap Eg(Q) and in the conduction band Ee(Q) are expressed as a function of the configurational coordinate Q of the system. Once the electron is excited into the conduction band, the charge state of the localized trap changes by +1 and significant lattice relaxation can occur. However, the Franck-Condon principle dictates that optical transitions occur without changes in the configuration and lattice relaxation only occurs once the electron is in its excited state. This is illustrated in Figure 2.8 by the vertical transition A→B, with energy Eo. After lattice relaxation, the excited electron loses an amount of energy Eph, transition B→C. Thermal transitions, on the other hand allow for lattice relaxations and the net energy required is shown as Et in the figure, with the transition A→C. Et and Eo are related as in Equation 2.13. Estimates reveal Eo/Et≈εo/ε, where εo and ε are the high frequency and static dielectric constants, respectively (Mott and Gurney 1948).

Figure 2.8 A configurational coordinate diagram showing the potential energy curves Eg(Q) and Ee(Q) in the region of the defect when the defect state is occupied by an electron, and when it is empty (ionized). When the level is occupied the energy is a minimum at configurational coordinate Qg. Optical transitions take place vertically (transition AB) since the lattice does not have time to respond to the change in charge state of the defect and relax to its new configurational coordinate. The optical energy required to affect this transition is Eo. Once ionized, the lattice relaxes to new coordinate Qe and a new energy minimum at C, following emission of phonons of energy Eph. Lattice relaxations are allowed during thermal excitations, however, and thermal stimulation can cause transitions directly from A to C. The required thermal energy is Et, where Et=Eo – Eph.

The form of the photoionization cross-section function σp(E) depends upon several material-related factors and different models exist to describe it. The first point to remember is that it is a function, and its value depends on the wavelength of the stimulating light used to excite the electrons from their traps. Thus, if broad-band, polychromatic excitation light is used, the photoionization cross-section will be multivalued. At first glance, one might expect that the function might be a step function since as long as the photon energy E=hv>Eo, trap ionization will occur, whereas if hv<Eo, there will be no excitation of the electron from the trap. However, the shape of the σp(E) function depends on the shape of the availability of states in the conduction band, i.e. the density of states function Z(E) and a peak in the value of σp is usually observed as the photon energy becomes too large for the corresponding states into which the electron could be excited. Furthermore, strong phonon coupling of the trapped electron to the lattice can give rise to excitation even if hν<Eo.

The various expressions for photoionization cross-section σp(E) depend upon assumptions relating to the potential energy in the vicinity of the defect, the wavefunctions for the trapped and delocalized states, the density of states in the delocalized band, and the degree of phonon interaction. For a shallow (hydrogenic) electron trap:

(2.15)

where E=hv is the energy of the stimulating light (Blakemore and Rahimi 1984; Landsberg 2003). The coulombic attraction between the freed electron and the ionized defect is ignored when hv is just larger than Eo. The cross-section reaches its maximum at hv=1.4 Eo.

For deep traps, Lucovsky (1964) approximated the potential in the region of the defect to a delta function and assumed a plane wave excited-state wavefunction to derive:

(2.16)

The cross-section reaches a maximum at hv=2Eo, and the coulombic field is taken into account.

A further assumption in the derivation of Equation (2.16) is that the effective mass me* of the electron in the conduction band can be used also for the electron in the localized state. By using the electron rest mass mo instead of me* while the electron is localized, Grimmeis and Ledebo (1975a, 1975b) derived:

(2.17)

also using a plane-wave final state and the assumption of parabolic bands.

By taking into account strong phonon coupling between the lattice and the trapped electron, Noras (1980) (see also Chruścińska 2010) derived:

(2.18)

The parameter ϵ is a dummy variable having the dimensions of energy, and a = 5/2 or 3/2 for forbidden and allowed transitions, respectively. The parameter κ is given by:

(2.19)

where again S is the Huang-Rhys factor and hvph is the energy of the phonon vibrational mode.

For a purely electronic transition (no phonon coupling):

(2.20)

for hv>>Eo, and σp(E)=0 for hv<Eo. Compare Equation 2.20 with Equations 2.16 and 2.17.

Several other expressions for σp(E) also exist (Jaros 1977; Blakemore and Rahimi 1984; Ridley 1988; Böer 1990; Landsberg 2003).

OSL signals from dosimetry materials originate from the release of electrons from deep trapping states and thus the expressions most frequently used to represent the photoionization cross-section of such centers are Equations 2.16, 2.17, and 2.18. A comparison of the shapes of some of these expressions is given in Figure 2.9.

Figure 2.009 (a) Examples of postulated photoionization cross-sections as a function of stimulation energy. In this depiction, all curves are normalized to their maximum value and the optical trap depth is Eo = 2.25 eV. (b) Example photoionization cross-sections when phonon coupling is allowed. In this figure, the Huang-Rhys factor S is 10 and the temperature is 300 K. Curves corresponding to two values for Eo are illustrated, each with two curve shapes corresponding to values of hvph of 20 meV (dashed lines) and 40 meV (full lines). (Adapted from Chrus´cin´ska 2010.)