Lattice defects in crystals

Positron lifetime measurements can be used to investigate the type and the density of lattice defects in crystals [293]. In solid materials positrons have a typical lifetime of 300 to 500 ps until they are annihilated by an electron. When positrons diffuse through a crystal they may be trapped in crystal imperfections. The electron density in these locations is different from the density in a defect-free crystal. Therefore, the positron lifetime depends on the type and the density of the crystal defects. When a positron annihilates with an electron two y quanta of 511 keV are emitted. The y quanta can easily be detected by a scintillator and a PMT. 

The variable properties of solids are coimected with the ability of molecules to exist in different states of order, ranging from closely packed molecular crystals with a minimum free energy to metastable crystal phases and, finally, to the glassy state with the highest free energy. This phenomenon is commonly referred to as polymorphism. Lattice defects in crystals and particularly solvate formation add another level of complexity. Whether a solid in any metastable state can be handled and analysed is a kinetic issue, which again is affected by many factors (e.g. chemical impurities, solvent residues, moisture, and interactions with drug excipients). 

An additional problem is encountered when the isolated solid is non-stoichiometric. For example, precipitating Mn + as Mn(OH)2, followed by heating to produce the oxide, frequently produces a solid with a stoichiometry of MnO ) where x varies between 1 and 2. In this case the nonstoichiometric product results from the formation of a mixture of several oxides that differ in the oxidation state of manganese. Other nonstoichiometric compounds form as a result of lattice defects in the crystal structure. ... 

Although several types of lattices have been described for ionic crystals and metals, it should be remembered that no crystal is perfect. The irregularities or defects in crystal structures are of two general types. The first type consists of defects that occur at specific sites in the lattice, and they are known as point defects. The second type of defect is a more general type that affects larger regions of the crystal. These are the extended defects or dislocations. Point defects will be discussed first. 

Lattice defects in silver halide crystals. Philos. Mag. 40, 667 (1949). 

Lattice defects in ionic crystals are interstitial ions and ion vacancies. In crystalline sodium chloride NaCl a cation vacancy Vn - is formed by producing a surface cation NaJ, (Nal - NaJ + Vua ) this is called the Schottky defect. On the other hand, in crystalline silver chloride AgCl a pair of cation vacancy Va,. and interstitial cation Ag is formed, (Ag - Agj + ) this is called the Frenkel... 

This expression is not strictly correct, only at absolute zero is this state realized. With an increase of temperature, however, the free energy of the system decreases with increasing entropy. Therefore at higher temperatures, the crystal develops lattice defects in both the metal and oxygen sites, known as Schottky defects (see Section 1.3.2). 

As mentioned above, the non-stoichiometric compounds originate from the existence of point defects in crystals. Let us consider a crystal consisting of mono-atoms. In ideal crystals of elements, atoms occupy the lattice points regularly. In real crystals, on the other hand, various kinds of point defects can exist in thermodynamic equilibrium. First, we shall consider vacancies , which are empty regular lattice points. Consider a crystal composed of one element which has N atoms sited on regular lattice points and vacancies,... 

The analytical formalism just discussed has two shortcomings first, the usage of quite particular hop length distribution and, secondly, the restriction to the steady-state properties. The Torrey model becomes inadequate for point defects in crystals, where single hop lengths A between the nearest lattice sites takes place, p(r) = <5(r - A) in equation (4.3.4). This results in the... 

Color Centers. Lattice defects in alkali halide crystals provide ideal trapping sites for electrons which in turn cause marked color changes in the system. Symons and Doyle (112) have reviewed the research on color centers in alkali halide crystals to about 1960. In... 

Point defects in crystal lattices can be classified into two essential types (Fig. 13.58) ... 

Fig. 13.5. Defects in crystal lattices (a) Frenkel defect (b) Schottky defect.

M. J. Norgett, AERE Harwell Report R7650, Atomic Energy Research Establishment, 1974. A General Formulation of the Problem of Calculating the Energies of Lattice Defects in Ionic Crystals. 

In a perfect crystal with periodic potentials, electron wave functions form delocalized Bloch waves [46]. Impurities and lattice defects in disordered... 

Iron (II) sulphide never has the precise composition FeS—the sulphur is always present in excess. This could be due either to the inclusion in the lattice of extra, interstitial S atoms or to the omission from it of some of the Fe atoms. The second explanation is correct (Hagg and Sucksdorff, 1933), the phenomcon being an example of lattice defect (p. 152). There are two types of lattice defect. In Schottky defects, found in iron(Il) sulphide, holes are left at random through the crystal because of migration of ions to the surface. In Frenkel defects, holes are left at random by atoms which have moved to interstitial positions. Silver bromide has a perfect face-centred cubic arrangement of Br ions but the Ag+ ions are partly in interstitial positions. The effect is even more marked in silver iodide (p. 153). 

Applications. We have performed, with the system described above, high-resolution, thermal-wave imaging on many different materials. We have detected and Imaged subsurface mechanical defects such as microcracks and voids, grain boundaries, grains, and dislocations, and dopant regions and lattice variations in crystals. 

The thermal conductivity of a-alumina single crystals as a function of temperature is given in Table 16 (from [2, 23]). Heat is conducted through a nonmetallic solid by lattice vibrations or phonons. The mean free path of the phonons determines the thermal conductivity and depends on the temperature, phonon-phonon interactions, and scattering from lattice defects in the solid. At temperatures below the low temperature maximum (below about 40°K), the mean free path is mainly determined by the sample size because of phonon scattering from the sample surfaces. Above the maximum, the... 

In crystals, impurities can take simple configurations. But depending on their concentration, diffusion coefficient, or chemical properties and also on the presence of different kind of impurities or of lattice defects, more complex situations can be found. Apart from indirect information like electrical measurements or X-ray diffraction, methods such as optical spectroscopy under uniaxial stress, electron spin resonance, channelling, positron annihilation or Extended X-ray Absorption Fine Structure (EXAFS) can provide more detailed results on the location and atomic structure of impurities and defects in crystals. Here, we describe the simplest atomic structures more complicated structures are discussed in other chapters. To explain the locations of the impurities and defects whose optical properties are discussed in this book, an account of the most common crystal structures mentioned is given in Appendix B. 

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