General relations between direct and reciprocal lattices

Koch, E.

doi:10.1107/97809553602060000572

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 1.1, pp. 2-3

Section 1.1.1. General relations between direct and reciprocal lattices

E. Koch^a

^a Institut für Mineralogie, Petrologie und Kristallographie, Universität Marburg, Hans-Meerwein-Strasse, D-35032 Marburg, Germany

1.1.1. General relations between direct and reciprocal lattices

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1.1.1.1. Primitive crystallographic bases

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The vectors a, b, c form a primitive crystallographic basis of the vector lattice L, if each translation vector $[{\bf t}\in {\bf L}]$ may be expressed as $[{\bf t}=u{\bf a}+v{\bf b}+w{\bf c}]$ with u, v, w being integers.

A primitive basis defines a primitive unit cell for a corresponding point lattice . Its volume V may be calculated as the mixed product (triple scalar product) of the three basis vectors: $[\eqalignno{V&=({\bf abc})={\bf a}\times{\bf b}\cdot{\bf c}\cr &=\left[\left|\matrix{a^2&ab\cos\gamma&ac\cos\beta\cr ab\cos\gamma&b^2&bc\cos\alpha\cr ac\cos\beta&bc\cos\alpha&c^2\cr}\right|\right]^{1/2}\cr &=abc[1-\cos^2\alpha-\cos^2\beta-\cos^2\gamma\cr &\quad+2\cos\alpha\cos\beta\cos\gamma]^{1/2}\cr &=2abc\bigg[\sin{\alpha+\beta+\gamma\over2}\sin{-\alpha+\beta+\gamma\over2}\cr &\quad\times\sin{\alpha-\beta+\gamma\over2}\sin{\alpha+\beta-\gamma\over2}\bigg]^{1/2}. &(1.1.1.1)}]$ Here a, b and c designate the lengths of the three basis vectors and $[\alpha={\bf b}\wedge{\bf c}]$ , $[\beta={\bf c}\wedge{\bf a}]$ and $[\gamma={\bf a}\wedge{\bf b}]$ the angles between them.

Each vector lattice L and each primitive crystallographic basis a, b, c is uniquely related to a reciprocal vector lattice $[{\bf L}^*]$ and a primitive reciprocal basis a*, b*, c*: $[\matrix{&\left.\eqalign{{\bf a}^*&={{\bf b}\times{\bf c}\over V}\quad{\rm or\quad}{\bf a}^*\cdot{\bf b}={\bf a}^*\cdot{\bf c}=0,\quad{\bf a}^*\cdot{\bf a}=1\semi\cr {\bf b}^*&={{\bf c}\times{\bf a}\over V}\quad{\rm or\quad}{\bf b}^*\cdot{\bf a}={\bf b}^*\cdot{\bf c}=0,\quad{\bf b}^*\cdot{\bf b}=1\semi\cr {\bf c}^*&={{\bf a}\times{\bf b}\over V}\quad{\rm or\quad}{\bf c}^*\cdot{\bf a}={\bf c}^*\cdot{\bf b}=0,\quad{\bf c}^*\cdot{\bf c}\,=1.}\right\}\quad (1.1.1.2)\cr\cr &{\bf L}^*=\{{\bf r}^*|{\bf r}^*=h{\bf a}^*+k{\bf b}^*+l{\bf c}^*\ {\rm and}\ h,k,l\ {\rm integers}\}.\hfill}]$ The lengths [a^*] , [b^*] and [c^*] of the reciprocal basis vectors and the angles $[\alpha^*={\bf b}^*\wedge{\bf c}^*]$ , $[\beta^*={\bf c}^*\wedge{\bf a}^*]$ and $[\gamma^*={\bf a}^*\wedge{\bf b}^*]$ are given by: $[\left.\matrix{a^*=\displaystyle{bc\sin\alpha\over V},\quad b^*={ac\sin\beta\over V},\quad c^*={ab\sin\gamma\over V},\cr\noalign{\vskip5.5pt} \cos\alpha^*=\displaystyle{\cos\beta\cos\gamma-\cos\alpha\over\sin\beta\sin\gamma},\cr \noalign{\vskip5.5pt}\cos\beta^*=\displaystyle{\cos\alpha\cos\gamma-\cos\beta\over\sin\alpha\sin\gamma},\cr\noalign{\vskip5.5pt} \cos\gamma^*=\displaystyle{\cos\alpha\cos\beta-\cos\gamma\over\sin\alpha\sin\beta}.}\right\}\eqno (1.1.1.3)]$ a*, b*, c* define a primitive unit cell in a corresponding reciprocal point lattice. Its volume V* may be expressed by analogy with V [equation (1.1.1.1)]: $[\eqalignno{V^*&=({\bf a}^*{\bf b}^*{\bf c}^*)={\bf a}^*\times{\bf b}^*\cdot{\bf c}^*\cr &=\left[\left|\matrix{a^{*2}&a^*b^*\cos\gamma^*&a^*c^*\cos\beta^*\cr a^*b^*\cos\gamma^*&b^{*2}&b^*c^*\cos\alpha^*\cr a^*c^*\cos\beta^*&b^*c^*\cos\alpha^*&c^{*2}}\right|\right]^{1/2}\cr &=a^*b^*c^*[1-\cos^2\alpha^*-\cos^2\beta^*-\cos^2\gamma^*\cr &\quad+2\cos\alpha^*\cos\beta^*\cos\gamma^*]^{1/2}\cr &=2a^*b^*c^*\left[\sin{\alpha^*+\beta^*+\gamma^*\over2}\sin{-\alpha^*+\beta^*+\gamma^*\over2}\right.\cr &\quad\times\left.\sin{\alpha^*-\beta^*+\gamma^*\over2}\sin{\alpha^*+\beta^*-\gamma^*\over2}\right]^{1/2}.& (1.1.1.4)}]$

In addition, the following equation holds: $[VV^*=1.\eqno (1.1.1.5)]$ As all relations between direct and reciprocal lattices are symmetrical, one can calculate a, b, c from a*, b*, c*: $[{\bf a}={{\bf b}^*\times{\bf c}^*\over V^*},\quad {\bf b}={{\bf c}^*\times{\bf a}^*\over V^*},\quad {\bf c}={{\bf a}^*\times{\bf b}^*\over V^*},\eqno (1.1.1.6)]$ $[\left.\matrix{a=\displaystyle{b^*c^*\sin\alpha^*\over V^*},\cr\noalign{\vskip5.5pt} b=\displaystyle{a^*c^*\sin\beta^*\over V^*},\cr\noalign{\vskip5.5pt} c=\displaystyle{a^*b^*\sin\gamma^*\over V^*},\cr\noalign{\vskip5.5pt} \cos\alpha=\displaystyle{\cos\beta^*\cos\gamma^*-\cos\alpha^*\over\sin\beta^*\sin \gamma^*},\cr\noalign{\vskip5.5pt}\cos\beta=\displaystyle{\cos\alpha^*\cos\gamma^*-\cos\beta^*\over \sin\alpha^*\sin\gamma^*},\cr \noalign{\vskip5.5pt}\cos\gamma=\displaystyle{\cos\alpha^*\cos\beta^*-\cos\gamma^*\over \sin\alpha^*\sin\beta^*}.}\right\}\eqno (1.1.1.7)]$ The unit-cell volumes V and V* may also be obtained from: $[\eqalignno{V&=abc\sin\alpha^*\sin\beta\sin\gamma\cr &=abc\sin\alpha\sin\beta^*\sin\gamma\cr &=abc\sin\alpha\sin\beta\sin\gamma^*,& (1.1.1.8)}]$ $[\eqalignno{V^*&=a^*b^*c^*\sin\alpha\sin\beta^*\sin\gamma^*\cr &=a^*b^*c^*\sin\alpha^*\sin\beta\sin\gamma^*\cr &=a^*b^*c^*\sin\alpha^*\sin\beta^*\sin\gamma.& (1.1.1.9)}]$

1.1.1.2. Non-primitive crystallographic bases

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For certain lattice types, it is usual in crystallography to refer to a `conventional' crystallographic basis $[{\bf a}_c,{\bf b}_c,{\bf c}_c]$ instead of a primitive basis a, b, c. In that case, $[{\bf a}_c]$ , $[{\bf b}_c]$ , and $[{\bf c}_c]$ with all their integral linear combinations are lattice vectors again, but there exist other lattice vectors $[{\bf t}\in{\bf L}]$ , $[{\bf t}=t_1{\bf a}_c+t_2{\bf b}_c+t_3{\bf c}_c,]$ with at least two of the coefficients [t_1] , [t_2] , [t_3] being fractional.

Such a conventional basis defines a conventional or centred unit cell for a corresponding point lattice, the volume [V_c] of which may be calculated by analogy with V by substituting $[{\bf a}_c,{\bf b}_c,{\bf c}_c]$ for a, b, and c in (1.1.1.1).

If m designates the number of centring lattice vectors t with $[0\le t_1,t_2,t_3\lt1]$ , [V_c] may be expressed as a multiple of the primitive unit-cell volume V: $[V_c=mV.\eqno (1.1.1.10)]$ With the aid of equations (1.1.1.2) and (1.1.1.3), the reciprocal basis $[{\bf a}^*_c,{\bf b}^*_c,{\bf c}^*_c]$ may be derived from $[{\bf a}_c,{\bf b}_c,{\bf c}_c]$ . Again, each reciprocal-lattice vector $[{\bf r}^*=h{\bf a}^*_c+k{\bf b}^*_c+l{\bf c}^*_c\in{\bf L}^*]$ is an integral linear combination of the reciprocal basis vectors, but in contrast to the use of a primitive basis only certain triplets h, k, l refer to reciprocal-lattice vectors.

Equation (1.1.1.5) also relates [V_c] to [V^*_c] , the reciprocal cell volume referred to $[{\bf a}^*_c,{\bf b}^*_c,{\bf c}^*_c]$ . From this it follows that $[V^*_c={1\over m}V^*.\eqno (1.1.1.11)]$

Table 1.1.1.1 contains detailed information on `centred lattices' described with respect to conventional basis systems.

Table 1.1.1.1| top | pdf |
Direct and reciprocal lattices described with respect to conventional basis systems

Direct lattice			Reciprocal lattice
$[{\bf a}_c, {\bf b}_c, {\bf c}_c]$			$[{\bf a}^_c, {\bf b}^_c, {\bf c}^*_c]$
Bravais letter	Centring vectors	Unit-cell volume	Conditions for reciprocal-lattice vectors $[h{\bf a}^_c+k{\bf b}^_c+l{\bf c}^*_c]$	Unit-cell volume	Bravais letter
A	$[{1\over2}{\bf b}_c+{1\over2}{\bf c}_c]$	2V		$[{1\over2}V^*]$	A
B	$[{1\over2}{\bf a}_c+{1\over2}{\bf c}_c]$	2V		$[{1\over2}V^*]$	B
C	$[{1\over2}{\bf a}_c+{1\over2}{\bf b}_c]$	2V		$[{1\over2}V^*]$	C
I	$[{1\over2}{\bf a}_c+{1\over2}{\bf b}_c+{1\over2}{\bf c}_c]$	2V		$[{1\over2}V^*]$	F
F	$[{1\over2}{\bf a}_c+{1\over2}{\bf b}_c,]$ $[{1\over2}{\bf a}_c+{1\over2}{\bf c}_c,]$ $[{1\over2}{\bf b}_c+{1\over2}{\bf c}_c]$	4V		$[{1\over4}V^*]$	I
R	$[{1\over3}{\bf a}_c+{2\over3}{\bf b}_c+{2\over3}{\bf c}_c]$ , $[{2\over3}{\bf a}_c+{1\over3}{\bf b}_c+{1\over3}{\bf c}_c]$	3V		$[{1\over3}V^*]$	R

As a direct lattice and its corresponding reciprocal lattice do not necessarily belong to the same type of Bravais lattices [IT A (2005, Section 8.2.5 )], the Bravais letter of $[{\bf L}^*]$ is given in the last column of Table 1.1.1.1. Except for P lattices, a conventionally chosen basis for $[{\bf L}^*]$ coincides neither with a*, b*, c* nor with $[{\bf a}^*_c,{\bf b}^*_c,{\bf c}^*_c]$ . This third basis, however, is not used in crystallography. The designation of scattering vectors and the indexing of Bragg reflections usually refers to $[{\bf a}^*_c,{\bf b}^*_c,{\bf c}^*_c]$ .

If the differences with respect to the coefficients of direct- and reciprocal-lattice vectors are disregarded, all other relations discussed in Part 1 are equally true for primitive bases and for conventional bases.

References

International Tables for Crystallography (2005). Vol. A, Space-group symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 1.1, pp. 2-3