View source for Reciprocal lattice

= reciprocal lattice =


=== Other languages ===

Réseau réciproque (''Fr''). Reziprokes Gitter (''Ge''). Red reciproca (''Sp'')

== Definition ==

The reciprocal lattice is constituted by the set of all possible linear combinations of the basis vectors 
'''a*''', '''b*''', '''c*''' of the [[reciprocal space]]. A point (''node''), ''H'', of the reciprocal lattice is defined by its position vector: 

'''OH''' = '''r<sub>hkl</sub>*''' = ''h'' '''a*''' + ''k'' '''b*''' + ''l'' '''c*'''.

The generalizaion of the reciprocal lattice in a four-dimensional space for incommensurate structures is described in Section 9.8 of ''International Tables of Crystallography, Volume C''.

== Geometrical applications ==

Each '''vector''' '''OH''' = '''r,,hkl,,*''' = ''h'' '''a*''' + ''k'' '''b*''' + ''l'' '''c*''' ''' of the reciprocal lattice is associated to a family of direct lattice planes'''. It is normal to the planes of the family, and the lattice spacing of the family is ''d,,hkl,,'' = 1/''n,,hkl,,''. If '''OP''' = ''x'' '''a''' + ''y'' '''b''' + ''z'' '''c''' is the position vector of a point of a lattice plane, the equation of the plane is given by '''OR''' . '''OP''' = ''K'' where ''K'' is a constant integer. Using the properties of the scalar product of a [[reciprocal space]] vector and a [[direct space]] vector, this equation is  
'''OR''' . '''OP''' = ''hx'' + ''ky'' + ''lz'' = ''K''. The Miller indices of the family are 
''h'', ''k'', ''l''.

The Miller indices of the '''family of lattice planes parallel to two direct space vectors''', 
'''r,,1,,''' = ''u,,1,,'' '''a''' + ''v,,1,,'' '''b''' + ''w,,1,,'' '''c''' and 
'''r,,2,,''' = ''u,,2,,'' '''a''' + ''v,,2,,'' '''b''' + ''w,,2,,'' '''c''' are proportional to the
coordinates in reciprocal space, ''h'', ''k'', ''l'', of the vector product of these two vectors:

''h''/(''v,,1,,'' ''w,,2,,'' - ''v,,2,,'' ''w,,1,,'') =
''k''/(''w,,1,,'' ''u,,2,,'' - ''w,,2,,'' ''u,,1,,'') =
''l''/(''u,,1,,'' ''v,,2,,'' - ''u,,2,,'' ''v,,1,,'').

The coordinates ''u'', ''v'', ''w'' in direct space of the '''zone axis intersection of two families of lattice planes''' of Miller indices ''h,,1,,'', ''k,,1,,'', ''l,,1,,'' and 
''h,,2,,'', ''k,,2,,'', ''l,,2,,'', respectively, are proportional to the coordinates of the vector product of the reciprocal lattice vectors associated to these two families:

''u''/(''k,,1,,'' ''l,,2,,'' - ''k,,2,,'' ''l,,1,,'') =
''v''/(''l,,1,,'' ''h,,2,,'' - ''l,,2,,'' ''h,,1,,'') =
''w''/(''h,,1,,'' ''k,,2,,'' - ''h,,2,,'' ''k,,1,,'').

== Diffraction condition in reciprocal space ==

The condition that the waves outgoing from two point scatterers separated by a lattice vector 
'''r''' = ''u'' '''a''' + ''v'' '''b''' + ''w'' '''c''' (''u'', ''v'', ''w'' integers) be in phase is that the scalar product ('''s<sub>h</sub>'''/&#955; - '''s<sub>o</sub>'''/&#955;) . '''r''', where '''s<sub>h</sub>''' and '''s<sub>o</sub>''' are unit vectors in the scattered and incident directions, respectively, be an integer, ''K''. This condition is satisfied if 
('''s<sub>h</sub>'''/&#955; - '''s<sub>o</sub>'''/&#955;) = ''h'' '''a*''' + ''k'' '''b*''' + ''l'' '''c*''', where ''h'', ''k'', ''l'' are integers, namely if the diffraction vector 
('''s<sub>h</sub>'''/&#955; - '''s<sub>o</sub>'''/&#955;) is a vector '''0H''' of the reciprocal lattice. A node of the reciprocal lattice is therefore associated to each Bragg reflection on the lattice
planes of Miller indices (''h/K'', ''k/K'', ''l/K''). It is called the ''hkl'' reflection. 

==== History ====

The notion of reciprocal vectors was introduced in vector analysis by J. W. Gibbs (1981 - ''Elements of Vector Analysis, arranged for the Use of Students in Physics''. Yale University, New Haven). The concept of reciprocal lattice was adapted by P. P. Ewald to interpret the diffraction pattern of an orthorhombic crystal (1913) in his famous paper where he introduced the sphere of diffraction. It was extended to lattices of any type of symmetry by M. von Laue 
(1914) and Ewald (1921). The first approach to that concept is that of the system of 
''polar axes'', introduced by Bravais in 1850, which associates the direction of its normal to a family of lattice planes. 

 


=== See also ===

[[reciprocal space]]

[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/4/ The Reciprocal Lattice]  (Teaching Pamphlet of the ''International Union of Crystallography'')

Section 5.1, ''International Tables of Crystallography, Volume A''

Section 1.1, ''International Tables of Crystallography, Volume B''

Section 1.1, ''International Tables of Crystallography, Volume C''

Section 1.1.2, ''International Tables of Crystallography, Volume D''