Difference between revisions of "Lattice"
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<Font color="blue">Réseau</Font>(''Fr''); <Font color="red">Gitter</Font> (''Ge''); <Font color="black">Reticolo</Font>(''It''); <font color="purple">格子</font> (''Ja''). | <Font color="blue">Réseau</Font>(''Fr''); <Font color="red">Gitter</Font> (''Ge''); <Font color="black">Reticolo</Font>(''It''); <font color="purple">格子</font> (''Ja''). | ||
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A '''lattice''' in the vector space '''V<sup>n</sup>''' is the set of all integral linear combinations '''t''' = ''u''<sub>1</sub>'''a<sub>1</sub>''' + ''u''<sub>2</sub>'''a<sub>2</sub>''' + ... + ''u''<sub>k</sub>'''a<sub>k</sub>''' of a system ('''a<sub>1</sub>''', '''a<sub>2</sub>''', ... , '''a<sub>k</sub>''') of linearly independent vectors in '''V<sup>n</sup>'''. | A '''lattice''' in the vector space '''V<sup>n</sup>''' is the set of all integral linear combinations '''t''' = ''u''<sub>1</sub>'''a<sub>1</sub>''' + ''u''<sub>2</sub>'''a<sub>2</sub>''' + ... + ''u''<sub>k</sub>'''a<sub>k</sub>''' of a system ('''a<sub>1</sub>''', '''a<sub>2</sub>''', ... , '''a<sub>k</sub>''') of linearly independent vectors in '''V<sup>n</sup>'''. | ||
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*[[crystallographic basis]]<br> | *[[crystallographic basis]]<br> | ||
*Sections 8.1 and 9.1 of ''International Tables for Crystallography, Volume A'' | *Sections 8.1 and 9.1 of ''International Tables for Crystallography, Volume A'' | ||
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[[Category:Fundamental crystallography]]<br> | [[Category:Fundamental crystallography]]<br> | ||
Revision as of 15:45, 6 February 2012
Réseau(Fr); Gitter (Ge); Reticolo(It); 格子 (Ja).
A lattice in the vector space Vn is the set of all integral linear combinations t = u1a1 + u2a2 + ... + ukak of a system (a1, a2, ... , ak) of linearly independent vectors in Vn.
If k = n, i.e. if the linearly independent system is a basis of Vn, the lattice is often called a full lattice. In crystallography, lattices are almost always full lattices, therefore the attribute "full" is usually suppressed.
See also
- crystallographic basis
- Sections 8.1 and 9.1 of International Tables for Crystallography, Volume A