Lattice defect grain boundary, solid solution

In order to establish the model of intergranular impedance for doped barium titanate, it is important to notice that miorostructure properties of BaTiOj based materials, expressed in their grain boundary contacts, are of basic importance for electric properties of these materials. The barrier character of the grain boundaries is especially pronounced for doped BaTiOs materials which are used as PTC resistors. Basically two types of dopants can be introduced into BaTiOs large ions of valence 3+ and higher, can be incorporated into Ba positions, while the small ions of valence 5+ and higher, can be incorporated into the Ti sublattice [9-11], Usually, the extent of the solid solution of a dopant ion in a host structure depends on the site where the dopant ion is incorporated into the host structure, the compensation mechanism and the solid solubility limit [12], For the rare-earth-ion incorporation into the BaTiOs lattice, the BaTiOs defect chemistry mainly depends on the lattice site where the ion is incorporated [13], It has been shown that the three-valent ions incorporated at the Ba -sites act as donors, which extra donor charge is compensated by ionized Ti vacancies (V -), the three-valent ions... 

The resistivity p = l/tr is found to have a linear relationship with temperature in the form p T) = po + aT (Matthiessen s rule), where po is due to collisions with structural and impurity imperfections and the temperature dependence comes about from collisions with the lattice ions whose cross sections increase linearly with temperature. Impurity atoms, such as foimd in solid solution alloys, produce a much larger increase in resistivity than structural defects such as dislocations and grain boundaries or condensed second phases because they are more widely dispersed. Also there is a departure from the linear temperature dependence of the resistivity at low temperatures because all of the phonon modes are active. Griineisen used the Debye theory to develop a universal relationship between reduced resistivity and reduced temperature that holds for all metals. 

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