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(See also: Crystal structure added)
 
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<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'').
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<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'').  
  
  
== Definition ==
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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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If ''k = n'', ''i.e.'' if the linearly independent system is a '''basis''' of '''V'''<sup>''n''</sup>, the lattice is often called a '''full lattice'''. In crystallography, lattices are almost always full lattices, therefore the attribute 'full' is usually suppressed.
 
 
If ''k = n'', i.e. if the linearly independent system is a '''basis''' of '''V<sup>n</sup>''', 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 ==
 
== See also ==
  
*[[crystallographic basis]]<br>
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*[[Crystallographic basis]]
*Sections 8.1 and 9.1 of ''International Tables of Crystallography, Volume A''
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*[[Crystal structure]]
 
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*Chapters 1.3.2 and 3.1 of ''International Tables for Crystallography, Volume A'', 6th edition
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[[Category:Fundamental crystallography]]<br>
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[[Category:Fundamental crystallography]]

Latest revision as of 15:23, 16 November 2018

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