affects the optical and electronic properties of the material and moving electronic charges also cause subsequent changes to the polarization and displacements (Hayes and Stoneham 1985).
Figure 2.2 (a) An idealized lattice for an ionic crystal of the type A+B–. (b) Stylized polarization effects caused by the substitution of an A+ ion with a divalent impurity ion X2+.
The schematic illustration in Figure 2.2b is for illustrative purposes only. Modern density functional theory (DFT) calculations are able to calculate the positions of the ions when a divalent impurity is introduced into an A+B– lattice. An example is shown in Figure 2.3 for LiF doped with Mg2+, with the Mg ion in an interstitial position. For another example, Masillon et al. (2019) reveal that when Mg2+ substitutes for Li+ there is a resultant displacement of six nearest F atoms symmetrically in pairs, by 0.14 Å, 0.15 Å and 0.18 Å, while one F atom close to the Li-vacancy moves by 0.15 Å. At the same time, three neighboring Li atoms move away from their original position; two by 0.11 Å and one by 0.14 Å.
Figure 2.3 Two views, (a) and (b), of density functional theory calculations for the distortions in the lattice caused by the addition of an interstitial Mg2+ ion impurity into LiF. Li+ ions are in grey; F– ions are in green, and the Mg2+ impurity ion is in orange. (Original data kindly provided by Guerda Masillon, © Guerda Masillon, UNAM, Mexico.)
The conclusion from these considerations is that a “point” defect in a lattice can exert influence over several lattice spacings and, in the certain cases, over several thousand surrounding host ions. Indeed, a “point defect” is not a “point” at all (Townsend 1992).
In a dilute system each defect can be considered “unseen” by other defects. In this definition, no matter over how many lattice spacings the defect can exert influence, there are no other defects within this sphere of influence. However, this is not always the case. For example, a divalent impurity substituting for an alkali ion can be charge compensated locally through coulombic attraction with an alkali vacancy. Using the popular TL dosimetry material LiF:Mg as an example, Mg2+ ions that substitute for host Li+ ions are charge compensated by Li vacancies, which are effectively negatively charged and occupy nearest-neighbor positions along the <110> direction in the lattice. The clustering process forms dipolar complexes (Figure 2.4a), which are revealed by DFT calculations in LiF:Mg (Masillon et al. 2019). However, at room temperature this process of dipole formation does not lead to the final thermodynamic equilibrium state of the system, and further reduction in the system’s free energy can be attained by further clustering of the dipoles to form trimers, consisting of three dipoles in one of several possible configurations, an example of which is shown in Figure 2.4b (Taylor and Lilley 1982a; McKeever 1985; Gavartin et al. 1991).
Figure 2.4 (a) Schematic view of a LiF lattice with Mg2+ impurity substituting for a Li+ host ion and charge compensated by a Li-vacancy in a <110> direction, forming a dipolar complex; (b) example trimer cluster of three Mg2+-Livac dipoles.
An additional consideration, not indicated in the conceptual Figures 2.2 and 2.4, is that the radius of the impurity ions generally do not match those of the host ions for which they substitute. For example, the radius of a Mg2+ ion is 86×10–12 m, whereas the radius of Li+ is 90×10–12 m. This small change in radius (<5%) causes a much larger change in ionic volume in that part of the lattice and substituting a Li+ ion with a Mg2+ ion results in a decrease of ionic volume by ~15%. Similarly, Ti4+ results in a ~76% volume decrease, while Y3+ in CaF2 causes a ~69% decrease when substituting for Ca2+. These effects immediately cause lattice distortions over and above distortions due to coulombic forces.1
Radiation-dosimetry-quality LiF does not contain only Mg impurities; the TL sensitivity of the material is strongly enhanced by the inclusion of Ti4+ impurities. Charge compensation in this case appears through the incorporation of OH– impurities, forming Ti-OH complexes. The role of the Ti-OH complexes in the production of TL is still not precisely determined but they are believed to be active in the recombination processes occurring in this material. In particular, much evidence has accumulated in recent years to suggest that the main TL peak from this material (the so-called peak 5) is the result of a Mg-trimer/Ti-OH complex of an as-yet-undetermined crystalline structure. That is, the Ti-OH complexes and the Mg-Livac trimers appear to be spatially associated in the LiF lattice and many of the characteristic TL properties of peak 5, and the TL glow curve in general, are uniquely affected by this association.
It is the breakdown in the lattice periodic potential caused by defects that gives rise to energy levels within the forbidden gap. The wavefunctions of electrons in a crystal with perfect periodicity are delocalized and extend throughout the material. States where the electron wavefunction is localized are not allowed. It is when the periodicity breaks down due to the presence of a defect that localized wavefunctions occur. These decay with distance away from the center of the defect over several lattice sites and the corresponding energies reside within the band gap.
Whether they are relatively simple defects or complex defects, interactions between the localized electrons and holes can take place either, or both, non-locally (i.e. via the delocalized bands) or locally (e.g. tunneling of charge between localized states) depending upon both the energy and the spatial association between the defects. The energy levels may be discrete (i.e. characterized by a single energy value) or, because of their complexity, can be distributed in energy with the exact energy value depending upon the nature of the surrounding environment and the presence of other defects. This is especially true of non-crystalline materials such as glasses, in which the surrounding lattice may display short- or long-range disorder, resulting in a range of energies for a particular defect type.
2.1.2 Extended Defects
In addition to point defects, one can also add even more complex defects consisting of line dislocations (boundaries between slipped and un-slipped lattice planes), grain boundaries, angular misfits between lattice planes, planar dislocations (internal surfaces), nanoparticle formations, inclusions, and precipitates. Another obvious cause of the breakdown in the periodicity of a lattice is the presence of a surface. Crystals are not infinitely large and at a surface the lattice periodicity ends abruptly giving rise to broken bonds and bonds passivated by the possible presence of foreign atoms, resulting in large concentrations of localized levels at the surface. Clearly, powdered materials with small grain sizes and large surface-to-volume ratios are more likely to exhibit effects due to surface states than larger, bulk materials.
2.1.3 Non-Crystalline Materials
An obvious example of a material in which the infinite periodicity of the lattice cannot be expected is a non-crystalline or glassy material. The amorphous nature of such materials means that, at best, only short-range order can be expected and once several lattice distances are considered the material cannot be defined by regular order. As a result, the presence of defects such as those described in the previous sections results in a range of potential energies to describe the defect, depending upon the local environment. Furthermore, the energy states available to free electrons and holes must be reconsidered
