2.4.2 Notations and their Use

	How do we treat point defects in perfect analogy to atoms and molecules in chemical reaction equations? A very clear way was suggested by Kröger and Vink, it is therefore called "Kröger-Vink notation" or "notation by structure elements".
		We define vacancies and interstitials as particles which occupy a defined site in a crystal and which may have a charge.
		Sites in a crystal are the points where the atoms, the interstitials, or the vacancies can be. For a crystal composed of two kinds of atoms we have, e.g., the "A-sites" and the "B-sites". An A-atom on an A-site we denote by AA, a vacancy on a B-site is a VB
		This leaves the interstitials out of the picture. We therefore simply name all possible interstitials sites with their own place symbol and write Ai or Bi for an A-atom or a B-atom, resp., on its approbriate interstitals site.
		An interstital site not occuopied by av interstitial atom then, by definition, is occupied by a vacancy and symbolized by Vi.
	In order to facilitate book keeping with respect to the electrical charge, we only note the excess charge relative to the neutral lattice. Positive excess charge is marked by a point (e.g. A·), negative charge by a dash (e.g. A´) to distinguish this relative charge from the absolute charge. If we consider a positively charged Na+ ion in the NaCl lattice, we write NaNa as long as it is sitting on its regular lattice position, i.e. without a charge symbol. If we now consider a vacancy on the Na-site, the Na-ion as interstitial, or a Ca++ ion on the Na-site, we write
		VA´, Nai, and CaNa· because this defines the charge relative to the neutral lattice.
		Running through all the possible combinations for our NaCl crystal with some Ca, we obtain the matrix

	This calls for a little exercise

	What have gained by this? We now can describe all kinds of structure elements - atoms, molcules and defects - and their reactions in a clear and unambigous way relative to the empty space. Let´s look at some examples
		Formation of Frenkel defects in , e.g., AgCl: AgAg + Vi=VAg´ + Agi·.
		We see, why we need the slightly strange construction of a vacancy on an interstitial site.
		Formation of Schottky defects for an AB crystal AA + BB=VA´ + VB· + AB
	This looks good. The question is, if we can use the mass action law to determine equibrium concentrations. If the Frenkel defect example could be seen as analogous to the chemical reaction A + B=AB, we could write a mass action law as follows:
		(¦AgAg¦ · ¦Vi¦)/ (¦VAg´¦ · ¦ Agi·¦)=const. (with the ¦A¦ meaning "concentration of A"). The reaction constant is a complex function of p and T and especially the chemical potentials of the particles involved.
	Unfortunately, this is wrong!
	Why? Well, the notion of chemical equilibrium and thus the mass action law, at the normal conditions of const. temperature T and pressure p stems from finding the minimum of the free enthalpie G (also called Gibbs energy). You may want to read up a bit on the concept of chemical equilibrium, this can be done in the link In other words, we are searching for the concentration values of the particles ni involved in the reaction, which, at a given temperature and pressure, lead to dG=0.
		dG=0 we can always write as a total differential with respect to the variables dni:
		dG=(dG/dn1)dn1 + (dG/dn2)dn2 + ..
		The partial derivatives are defined as the chemical potentials of the particles in question and we always have to keep in mind that the long version of the above equation has a subscript at the partial derivative, which we, like many others, conviniently ""forgot" in the above equation. If written correctly (albeit, in html, somewhat akwardly), the partial derivative for the particle ni reads:
		(dG/dni)p,T,nj=not i, meaning that T, p, and all other particle concentrations must be kept constant.
	Only if that condition is fulfilled, can we formulate a mass action equation involving all particles present in the reaction equation!
		This "independence condition" is automatically not fulfilled if we have additional constraints which link some of our particles. And such constraints we do have in the Kröger-Vink notation!
		There is no way within the system to produce a vacany, e.g. VA without removing a A-particle, e.g. generating a Ai or adding another B-particle, BB.
	We now have a very useful way of describing chemical reactions, including all kinds of charged defects, but we cannot use simple thermodynamics. That´s where other notations come in
	You now may ask: Why not introduce the one notation that has it all and be done with it? The answer is: It could be done, but only by loosing simplicity in describing reactions. And simplicity is what you need in real (research) life, when you don´t know what is going on, and you try to get an answer by mulling over various possibility in your mind or on a sheet of paper.
	So "defects-in-ceramics" people live with several kinds of notation, all having pro and cons, and, after finding a good formulation in one notation, translate it to some other notation to get the answers required. We will provide a glimpse of this in the next subchapter.