### 2.4.3 Schottky Notation and Working with various notations

 The Kröger-Vink notation defined structure elements - atoms, molecules, point defects and even electrons and holes relative to empy space. Despite the problem with the unapplicability of the mass action law, this notation is in use throughout the scientific community dealing with point defects. The other important notation - the "Schottky notation" or "building element notation" is defined as follows: Defects are defined relative to the perfect crystal Charges then are automatically notated as in the Kröger-Vink way, i.e. relative to the perfect crystal. We again use the · for positive (relative) charge and the ´ for (relative) negative charge. To make things a bit more complicated, there are two ways of writing the required symbols, the "old" and the "new" Schottky notation. The "old" Schottky notation used special graphical symbols, like black circles or squares which are not available in html anyway. So we only give the new Schottky notation in direct comparison with the Kröger-Vink notation, again for the example NaCl with Ca impurities, i.e. A=Na+, B=Cl-, C=Ca+*
 A on B site A-vacancy A-interstitial Schottky (new) Na¦Cl¦·· ¦Na¦´ Na· Kröger-Vink NaCl·· VNa´ Nai· The "¦" sign is in the proper notation one vertical line; here we use whatever comes close in html. Do not confuse it with a symbol for concentration as used before
 So far, the difference between the Schottky notation and the Kröger-Vink notation seems superficial. The important difference, however, becomes clear upon writing down defect reactions. Lets look at the formation of Frenkel and Schottky defects in AgCl in the two notations.
 Frenkel defects Schottky defects Schottky (new) ¦Ag¦´ + Ag·=0 ¦Ag¦´ + ¦Cl¦·+ AB=0 Kröger-Vink AgAg + Vi=VAg´ + Agi·. AgAg + ClCl=VAg´ + VCl· + AB
 In words, the Schottky notation says: A (neg. charged) Ag-vacancy plus a (positively charged) Ag interstital gives zero for Frenkel defects A (neg. charged) Ag-vacancy plus a (positively charged) Cl vacancy plus a AgCl "lattice molecule" gives zero for Schottky defects This is clear enough for these simple cases, but not as clear and easy as the Kröger-Vink notation. But, and that is the big advantage, we can apply the mass action law directly to the reactions in the Schottky notation. This is not directly obvious. After all, Frenkel defects, e.g., do not only appear to be linked (where there is an interstitial, there is also a vacancy), but actually are linked if the defects are charged (otherwise there would be either net charge in the crystal or we would have to invoke electrons or holes to compensate the ionic charge an would have to include those into the recation equation). Theoretically, however, you can introduce one more vacancy or one more interstitial into a crystal with a given concentration of each and look at the change of the free enthalpy, i.e. the chemical potential of the species under consideration. If you do neglect the energy associated with charge (i.e you look at the chemical and not the electrochemical potential), the answers you get will not contain the coupling between the defects and you have to consider that separately. We will see how this works right away. Now, why dont we use just the Schottky notation and forget about Kröger-Vink? We asked that question before; the answer hasn´t changed: If we look at more complicatd reactions, e.g. between point defects in an ionic crystal, a gas on its outside, and electrons and holes for compensating charges, it os much easier to formulate possible reaction in the Kröger-Vink notation. The trick now is, to convert your reaction equations from the Kröger-Vink structure elements to the Schottky building elements. There is a simple recipe for doing this.
 All we have to do, is to combine the two structure elements of Kröger-Vink that refer to the same place in the lattice and view the combination as a building element. Let´s first look at an example and the generalize. Consider the Frenkel disorder in AgCl . Using structure element, we write AgAg + Vi=VAg´ + Agi·. Combining the terms referring to the same place in the lattice with the acual defects always as the first term in the combination yields (VAg´ - AgAg) + (Agi· - Vi)=0 Now all we have do is to write down the corresponding Schottky notation and identify the terms in brackets with the Schottky structure elements. We obtain the (VAg´ - AgAg)=¦Ag¦´ (Agi· - Vi)=Ag·. We can generalize this into a "translation" table
 A on B site A-vacancy A-interstitial AB molecule C on A site free electron hole All defects neutral Schottky (new) Building elements A¦B¦ ¦A¦ A AB C¦B¦ e´ h· Kröger-Vink Structure elements AB VA Ai AB CB e´ h· Kröger-Vink Combined structure elements= Building elements AB - BB VA - AA Ai - Vi AB CB - BB e´ h· The "¦" sign is in the proper notation one vertical line; here we use whatever comes close in html. Do not confuse it with a symbol for concentration as used before
 Let´s look at an example of how this works