A Course in Luminescence Measurements and Analyses for Radiation Dosimetry. Stephen W. S. McKeever

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      Figure 2.5 Density of states functions Z(E) for (a) a crystalline solid, and (b) an amorphous (non-crystalline) solid. For non-crystalline materials the density of states extends into the band gap (band-tail states) and localized states (shaded) are distributed in energy.

      It should also be mentioned that even crystalline materials can exhibit band-tail states extending into the gap if they exhibit sufficient lattice disorder due to variations in bond lengths, bond angles, local density fluctuations, and/or contain high concentrations of impurities or other extrinsic defects. If the disorder is sufficiently high, it can give rise to a density of states Z(E) that extends into the band gap. Such materials may include, for example, natural minerals, such as the feldspar family or the various polymorphs of silicon dioxide.

      2.2 Trapping, Detrapping, and Recombination Processes

      2.2.1 Excitation Probabilities

      2.2.1.1 Thermal Excitation

      Consider an electron localized at a lattice defect, at energy Et below the conduction band and probability p that the electron will absorb external energy and be excited from the trap into the conduction band. If the temperature of the system is T, the probability per second p that the electron will be thermally excited into the conduction band is given by:

(2.1)

      where v is the lattice phonon vibration frequency, K is the transition probability constant, k is Boltzmann’s constant, F is the Helmholtz free energy = Et−ΔST, and ΔS is the entropy change associated with the transition. Thus:

       (2.2)

      Here, s is known as the “attempt-to-escape” frequency (also known as the “frequency factor,” or the “pre-exponential factor”), with units of s^–1. It is the number of times per second that energy is absorbed from phonons in the lattice, and the term exp{−EtkT} is the probability that the energy absorbed is enough to cause a transition from the localized state to the conduction band. Typically, one can expect s ~ 10¹²–10¹⁴ s^–1; that is, of the order of the lattice vibration frequency, v, but differing from it by the factor Kexp{ΔSk}.

      By applying equilibrium statistics, it is possible to show that:

(2.3)

      where, Nc is the concentration of available states in the conduction band (units, m^–3), ve is the thermal velocity of free electrons (m.s^–1) and σ is the capture cross-section for the trap (m²).

      The concentration of free electrons, nc at any given temperature may be written:

       (2.4)

      Since the occupancy of the conduction band is essentially zero at the top of the conduction band, the integral can be taken to infinity. It is also assumed that Ec – EF >> kT, which is a good approximation in insulators, even for T = 1000s K. Nc can, therefore, be considered to be the effective density of states of a fictional level lying at the conduction band edge and is defined by:

(2.5)

      Similarly, the number of free holes in the valence band is given by:

       (2.6)

      and

(2.7)

      is the density of available states in the valence band. In Equations 2.5 and 2.7, me* and mh* are the effective masses of the free electrons and free holes in the conduction and valence bands, respectively, and h is Planck’s constant.

The value of the capture cross-section σ is critically dependent upon the potential distribution in the neighborhood of the trap and, in particular, upon whether the trap is coulombic attractive, neutral, or repulsive. Three representative cases are illustrated in Figure 2.6. The figures show the potential distribution around a coulombic attractive trap (a), a coulombic neutral trap (b), and a coulombic repulsive trap (c). The critical distance rc is that distance for which the energy of coulombic attraction equals the kinetic energy KE of the free electron (Rose 1963). If the trap has a coulombic net charge of +1, then: