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

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Название A Course in Luminescence Measurements and Analyses for Radiation Dosimetry
Автор произведения Stephen W. S. McKeever
Жанр Физика
Серия
Издательство Физика
Год выпуска 0
isbn 9781119646921



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applying equilibrium statistics, it is possible to show that:

      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:

      n Subscript c Baseline equals integral Subscript upper E Subscript c Baseline Superscript infinity Baseline upper Z left-parenthesis upper E right-parenthesis f left-parenthesis upper E right-parenthesis d upper E almost-equals upper N Subscript c Baseline exp left-brace minus StartFraction upper E Subscript c Baseline minus upper E Subscript upper F Baseline Over k upper T EndFraction right-brace period (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:

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

      m Subscript v Baseline almost-equals upper M Subscript v Baseline exp left-brace minus StartFraction upper E Subscript upper F Baseline minus upper E Subscript v Baseline Over k upper T EndFraction right-brace (2.6)

      and

      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.

      Figure 2.6 Potential ϕ as a function of distance r in the vicinity of: (a) coulombic attractive, (b) neutral, and (c) repulsive localized states. The critical distance rc, where σ=πrc2, is defined in the text. For coulombic repulsive centers the capture cross-section σ is reduced exponentially as a function of the potential barrier Δϕ.

      upper K upper E equals StartFraction q squared Over r Subscript c Baseline epsilon EndFraction comma (2.8)

      where, q is the charge on the electron and ε is the dielectric constant of the material. From this:

      sigma equals pi r Subscript c Superscript 2 Baseline equals pi left-parenthesis StartFraction q squared Over upper K upper E epsilon EndFraction right-parenthesis squared period (2.9)

      For coulombic repulsive centers the capture cross-section σ is reduced exponentially as a function of the potential barrier Δϕ – i.e. by exp{−qΔϕkT}. Thus, the capture cross-section of a repulsive trap is exponentially dependent on temperature.

      Experimentally determined values for σ range from ~10–16 m2 for attractive centers (so-called giant traps) to ~10–26 m2 for repulsive centers. Also, since ve∝T1/2, then KE∝T, and σ∝T−2 for attractive or neutral centers. It is clear that a coulombic attractive trap for a free electron is a repulsive trap for a free hole, and vice-versa. Thus, each localized state is represented by two cross-sections, one for electrons, σe and one for holes, σh. If σe>>σh the state is defined as an electron trap, while for a hole trap σh>>σe.

      Apart from thermal excitation out of the localized state, there is also the possibility that the trapped electron might attract an oppositely charged hole and the two may recombine. If a recombination event is more likely that a detrapping event, the localized state is called a recombination center. If the opposite is true, it is a trap. Thus, one can imagine that at a given temperature T, there may exist a state for which the probabilities are equal, that is:

      for electrons, and

      Also to be noted is that Equations 2.10 and