Vegard's rule

The Vegard'sche rule describes the linear dependence of the lattice constant of a substitutional solid solution or an alloy of the percentage of the components.

For a mixed crystal , consisting of the two components A and B, it can be formulated mathematically as follows:

{\ displaystyle a_ {AB} = x_ {A} \ cdot a_ {A} + (1-x_ {A}) \ cdot a_ {B}}

With

${\ displaystyle a_ {AB}}$ the lattice constant of the resulting mixed crystal
${\ displaystyle a_ {A}}$ and the lattice constants of the pure components ${\ displaystyle a_ {B}}$
${\ displaystyle x_ {A}}$ the percentage content of component A.

The prerequisite is that both components have the same structure type . The difference in the atomic or ionic radii is usually less than 15%.

This rule is especially used for alloys. The lattice constant of the alloy decreases with a smaller atomic volume of the alloying element and increases with a larger atomic volume. For example, adding tantalum or zirconium to titanium leads to an increase in the lattice constants, while increasing the levels of vanadium or molybdenum decrease them.

Although Vegard's rule is not exactly fulfilled in many cases, it serves as a guideline for the investigation of mixed crystals. Deviations from Vegard's rule are often examined.

For example, the lattice constants in alloys are slightly above the Vegard line if the binding forces between different types of atoms are smaller than between the same types of atoms (e.g. the alloys Cu-Au and Cu-Pd); in the opposite case, the lattice constants are below this straight line (e.g. B. the alloys Ag-Au and Ag-Pd).

literature

Will Kleber , Hans-Joachim Bautsch , Joachim Bohm , Detlef Klimm: Introduction to crystallography . 19th edition. Oldenbourg Wissenschaftsverlag, Munich 2010, ISBN 978-3-486-59075-3 .

L. Vegard: The constitution of mixed crystals and the space filling of the atoms . In: Journal of Physics . 5, No. 1, 1921, pp. 17-26. doi : 10.1007 / BF01349680 .

Individual evidence

↑ L. Vegard : The constitution of mixed crystals and the space filling of the atoms . In: Journal of Physics . 5, No. 1, 1921, pp. 17-26. bibcode : 1921ZPhy .... 5 ... 17V . doi : 10.1007 / BF01349680 .
^ AR Denton, NW Ashcroft: Vegard's law . In: Phys. Rev. A . 43, No. 6, 1991, pp. 3161-3164. bibcode : 1991PhRvA..43.3161D . doi : 10.1103 / PhysRevA.43.3161 .

[vegard21-1] L. Vegard : The constitution of mixed crystals and the space filling of the atoms . In: Journal of Physics . 5, No. 1, 1921, pp. 17-26. bibcode : 1921ZPhy .... 5 ... 17V . doi : 10.1007 / BF01349680 .

[denton91-2] AR Denton, NW Ashcroft: Vegard's law . In: Phys. Rev. A . 43, No. 6, 1991, pp. 3161-3164. bibcode : 1991PhRvA..43.3161D . doi : 10.1103 / PhysRevA.43.3161 .