Lattice

From Online Dictionary of Crystallography

(Difference between revisions)
Jump to: navigation, search
m (Style edits to align with printed edition)
m (languages)
Line 1: Line 1:
-
<Font color="blue">R&eacute;seau</Font>(''Fr''). <Font color="red">Gitter</Font> (''Ge''). <Font color="black">Reticolo</Font>(''It''). <font color="purple">格子</font> (''Ja'').
+
<Font color="orange">مشبك</Font> (''Ar''); <Font color="blue">R&eacute;seau</Font> (''Fr''); <Font color="red">Gitter</Font> (''Ge''); <Font color="black">Reticolo</Font> (''It''); <font color="purple">格子</font> (''Ja''); <font color="brown">Решётка</font> (''Ru''); <Font color="green">Red</Font> (''Sp'').  
 +
 
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