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International
Tables for Crystallography Volume A Space-group symmetry Edited by Th. Hahn © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. A. ch. 9.2, p. 750
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A primitive basis a, b, c is called a `reduced basis' if it is right-handed and if the components of the metric tensor G (cf. Chapter 9.1![[link]](/graphics/greenarr.gif)
) ![[\matrix{{\bf a \cdot a} &{\bf b \cdot b} &{\bf c \cdot c}\cr {\bf b \cdot c} &{\bf c \cdot a} &{\bf a \cdot b}\cr} \eqno(9.2.2.1)]](/teximages/abch9o2/abch9o2fd1.svg)
satisfy the conditions shown below. The matrix (9.2.2.1) for the reduced basis is called the reduced form.
Because of lattice symmetry there can be two or more possible orientations of the reduced basis in a given lattice but, apart from orientation, the reduced basis is unique.
Any basis, reduced or not, determines a unit cell – that is, the parallelepiped of which the basis vectors are edges. In order to test whether a given basis is the reduced one, it is convenient first to find the `type' of the corresponding unit cell. The type of a cell depends on the sign of ![[T = ({\bf a \cdot b}) ({\bf b \cdot c}) ({\bf c \cdot a}).]](/teximages/abch9o2/abch9o2fd2.svg)
If ![[T \gt 0]](/teximages/abch9o2/abch9o2fi1.svg)
, the cell is of type I, if ![[T \leq 0]](/teximages/abch9o2/abch9o2fi2.svg)
it is of type II. `Type' is a property of the cell since T keeps its value when a, b or c is inverted. Geometrically speaking, such an inversion corresponds to moving the origin of the basis towards another corner of the cell. Corners with all three angles acute occur for cells of type I (one opposite pair, the remaining six corners having one acute and two obtuse angles). The other alternative, specified by main condition (ii) of Section 9.2.3, viz all three angles non-acute, occurs for cells of type II (one or more opposite pairs, the remaining corners having either one or two acute angles).
The conditions can all be stated analytically in terms of the components (9.2.2.1), as follows:
![[\eqalignno{{\bf a \cdot a} \leq {\bf b \cdot b} \leq {\bf c \cdot c}\semi \quad& |{\bf b \cdot c}| \leq {\textstyle{1 \over 2}} {\bf b \cdot b}\semi \quad |{\bf a \cdot c}| \leq {\textstyle{1 \over 2}} {\bf a \cdot a}\semi & \cr & |{\bf a \cdot b}| \leq {\textstyle{1 \over 2}} {\bf a \cdot a} &(9.2.2.2a)\cr {\bf b \cdot c} \;\gt\; 0\semi \quad& \phantom{|}{\bf a \cdot c} \;\gt\; 0\semi \quad{\bf a \cdot b} \;\gt\; 0. & (9.2.2.2b)}]](/teximages/abch9o2/abch9o2fd3.svg)
![[\eqalignno{&\hbox{if } {\bf a \cdot a} = {\bf b \cdot b}\quad\hbox{then}\quad {\bf b \cdot c} \leq {\bf a \cdot c} & (9.2.2.3a) \cr &\hbox{if } {\bf b \cdot b} = {\bf c \cdot c}\quad \hbox{then}\quad {\bf a \cdot c} \leq {\bf a \cdot b} & (9.2.2.3b) \cr &\hbox{if } {\bf b \cdot c} = \textstyle{{1 \over 2}} {\bf b \cdot b}\quad\hbox{then}\quad{\bf a \cdot b} \leq 2{\bf a \cdot c}& (9.2.2.3c)\cr &\hbox{if } {\bf a \cdot c} = \textstyle{{1 \over 2}} {\bf a \cdot a}\quad\hbox{then}\quad{\bf a \cdot b} \leq 2{\bf b \cdot c}& (9.2.2.3d)\cr &\hbox{if } {\bf a \cdot b} = \textstyle{{1 \over 2}} {\bf a \cdot a}\quad\hbox{then}\quad {\bf a \cdot c} \leq 2{\bf b \cdot c.} & (9.2.2.3e)}]](/teximages/abch9o2/abch9o2fd4.svg)
![[\displaylines{\hfill \hbox{as } (9.2.2.2a) \hfill (9.2.2.4a)\cr \hfill (|{\bf b \cdot c}| + |{\bf a \cdot c}| + |{\bf a \cdot b}|) \leq {\textstyle{1 \over 2}} ({\bf a \cdot a} + {\bf b \cdot b}) \hfill (9.2.2.4b)\cr \hfill {\bf b \cdot c} \leq 0; \ \qquad {\bf a \cdot c} \leq 0; \ \qquad {\bf a \cdot b} \leq 0. \hfill (9.2.2.4c)}]](/teximages/abch9o2/abch9o2fd5.svg)
![[\eqalignno{&\hbox{if }{\bf a \cdot a} = {\bf b \cdot b}\quad\hbox{then}\quad |{\bf b \cdot c}| \leq |{\bf a \cdot c}| &(9.2.2.5a)\cr & \hbox{if } {\bf b \cdot b} = {\bf c \cdot c} \quad \hbox{then}\quad |{\bf a \cdot c}| \leq |{\bf a \cdot b}| & (9.2.2.5b)\cr & \hbox{if } |{\bf b \cdot c}| = {\textstyle{1 \over 2}}{\bf b \cdot b}\quad \hbox{then}\quad {\bf a \cdot b} = 0 & (9.2.2.5c)\cr & \hbox{if } |{\bf a \cdot c}| = {\textstyle{1 \over 2}}{\bf a \cdot a}\quad \hbox{then}\quad{\bf a \cdot b} = 0 & (9.2.2.5d)\cr& \hbox{if } |{\bf a \cdot b}| = {\textstyle{1 \over 2}}{\bf a \cdot a}\quad \hbox{then}\quad{\bf a \cdot c} = 0& (9.2.2.5e)\cr & \hbox{if } (|{\bf b \cdot c}| + |{\bf a \cdot c}| + |{\bf a \cdot b}|) = {\textstyle{1 \over 2}}({\bf a \cdot a} + {\bf b \cdot b})&\cr &\quad\hbox{then}\quad{\bf a \cdot a} \leq 2|{\bf a \cdot c}| + |{\bf a \cdot b}|.&(9.2.2.5f)\cr}]](/teximages/abch9o2/abch9o2fd6.svg)
The geometrical interpretation in the following sections is given in order to make the above conditions more explicit rather than to replace them, since the analytical form is obviously the most suitable one for actual verification.
