Difference between revisions of "Lattice"
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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>. | ||
Revision as of 09:21, 12 October 2017
مشبك (Ar); Réseau (Fr); Gitter (Ge); Reticolo (It); 格子 (Ja); Решётка (Ru); Red (Sp).
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
- Chapters 1.3.2 and 3.1 of International Tables for Crystallography, Volume A, 6th edition