Goldschmidt tolerance factor
{{short description|Factor used to determine the compatibility of an ion with a crystal structure}}
Goldschmidt's tolerance factor (from the German word Toleranzfaktor) is an indicator for the stability and distortion of crystal structures.{{cite book |url=https://books.google.com/books?id=N-rXAAAAMAAJ&q=Handbook+of+magnetism+and+advanced+magnetic+materials |last1=Parkin|first1=((editors-in-chief, Helmut Kronmller, Stuart))|title=Handbook of magnetism and advanced magnetic materials|year=2007|publisher=John Wiley & Sons|location=Hoboken, NJ|isbn=978-0-470-02217-7|author2=Mats Johnsson|edition=[Online-Ausg.]|author3=Peter Lemmens|accessdate=17 May 2012}} It was originally only used to describe the perovskite ABO3 structure, but now tolerance factors are also used for ilmenite.{{cite journal |last1=Liu |first1=XiangChun |author2=Hong, Rongzi; Tian, Changsheng |title=Tolerance factor and the stability discussion of ABO3-type ilmenite |journal=Journal of Materials Science: Materials in Electronics |date=24 April 2008 |volume=20 |issue=4 |pages=323–327 |doi=10.1007/s10854-008-9728-8 |s2cid=96085518 }}
Alternatively the tolerance factor can be used to calculate the compatibility of an ion with a crystal structure.{{cite web |last=Schinzer |first=Carsten |title=Distortion of Perovskites |url=http://www.ccp14.ac.uk/ccp/web-mirrors/pki/uni/pki/members/schinzer/stru_chem/perov/di_gold.html |accessdate=17 May 2012}}
The first description of the tolerance factor for perovskite was made by Victor Moritz Goldschmidt in 1926.{{cite journal|last=Goldschmidt|first=Victor M.|title=Die Gesetze der Krystallochemie|journal=Die Naturwissenschaften|year=1926|volume=14|issue=21|pages=477–485|doi=10.1007/bf01507527|bibcode=1926NW.....14..477G|s2cid=33792511}}
Mathematical expression
The Goldschmidt tolerance factor () is a dimensionless number that is calculated from the ratio of the ionic radii:
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| colspan="3" align="center" | | ||
| rA is the radius of the A cation. | rB is the radius of the B cation. | rO is the radius of the anion (usually oxygen). |
In an ideal cubic perovskite structure, the lattice parameter (i.e., length) of the unit cell (a) can be calculated using the following equation:
| style=" background: #fdfdfd; border:1px solid #ddd; text-align:center; margin:auto" cellspacing="15" | ||
| colspan="3" align="center" | | ||
| rA is the radius of the A cation. | rB is the radius of the B cation. | rO is the radius of the anion (usually oxygen). |
Perovskite structure
The perovskite structure has the following tolerance factors (t):
| class="wikitable" | ||||
| Goldschmidt tolerance factor (t) | Structure | Explanation | Example | Example lattice |
|---|---|---|---|---|
| >1 | Hexagonal or Tetragonal | A ion too big or B ion too small.
|
| - | ||
| 0.9-1 | Cubic | A and B ions have ideal size.
| | ||
| 0.71 - 0.9 | Orthorhombic/Rhombohedral | A ions too small to fit into B ion interstices.
|
| ||
| <0.71 | Different structures | A ions and B have similar ionic radii.
| | - |
