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

International Tables for Crystallography (2006). Vol. C, ch. 1.1, pp. 3-4

Section 1.1.2. Lattice vectors, point rows, and net planes

E. Kocha

aInstitut für Mineralogie, Petrologie und Kristallographie, Universität Marburg, Hans-Meerwein-Strasse, D-35032 Marburg, Germany

1.1.2. Lattice vectors, point rows, and net planes

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The length t of a vector [{\bf t}=u{\bf a}+v{\bf b}+w{\bf c}] is given by [\eqalignno{t^2&=u^2{\bf a}^2+v^2{\bf b}^2+w^2{\bf c}^2+2uvab\cos\gamma \cr&\quad+2uwac\cos\beta+2vwbc\cos\alpha.&(1.1.2.1)}]Accordingly, the length [r^*] of a reciprocal-lattice vector [{\bf r}^*= h{\bf a}^*+k{\bf b}^*+l{\bf c}^*] may be calculated from [\eqalignno{r^{*2}&=h^2a^{*2}+k^2b^{*2}+l^2c^{*2}+2hka^*b^*\cos\gamma^*\cr&\quad+2hla^*c^*\cos\beta^*+2klb^*c^*\cos\alpha^*.&(1.1.2.2)}]If the coefficients u, v, w of a vector [{\bf t}\in{\bf L}] are coprime, [uvw] symbolizes the direction parallel to t. In particular, [uvw] is used to designate a crystal edge, a zone axis, or a point row with that direction.

The integer coefficients h, k, l of a vector [{\bf r}^*\in{\bf L}^*] are also the coordinates of a point of the corresponding reciprocal lattice and designate the Bragg reflection with scattering vector r*. If h, k, l are coprime, the direction parallel to r* is symbolized by [[hkl]^*].

Each vector r* is perpendicular to a family of equidistant parallel nets within a corresponding direct point lattice. If the coefficients h, k, l of r* are coprime, the symbol (hkl) describes that family of nets. The distance d(hkl) between two neighbouring nets is given by [d(hkl)=r^{*-1}.\eqno (1.1.2.3)]Parallel to such a family of nets, there may be a face or a cleavage plane of a crystal.

The net planes (hkl) obey the equation [hx+ky+lz=n\quad(n={\rm integer}).\eqno (1.1.2.4)]Different values of n distinguish between the individual nets of the family; x, y, z are the coordinates of points on the net planes (not necessarily of lattice points). They are expressed in units a, b, and c, respectively.

Similarly, each vector [{\bf t}\in{\bf L}] with coprime coefficients u, v, w is perpendicular to a family of equidistant parallel nets within a corresponding reciprocal point lattice. This family of nets may be symbolized [(uvw)^*]. The distance [d^*(uvw)] between two neighbouring nets can be calculated from [d^*(uvw)=t^{-1}.\eqno (1.1.2.5)]A layer line on a rotation pattern or a Weissenberg photograph with rotation axis [uvw] corresponds to one such net of the family [(uvw)^*] of the reciprocal lattice.

The nets [(uvw)^*] obey the equation [uh+vk+wl=n \quad (n={\rm integer}).\eqno (1.1.2.6)]Equations (1.1.2.6)[link] and (1.1.2.4)[link] are essentially the same, but may be interpreted differently. Again, n distinguishes between the individual nets out of the family [(uvw)^*]. h, k, l are the coordinates of the reciprocal-lattice points, expressed in units [a^*], [b^*], [c^*], respectively.

A family of nets (hkl) and a point row with direction [uvw] out of the same point lattice are parallel if and only if the following equation is satisfied: [hu+kv+lw=0.\eqno (1.1.2.7)]

This equation is called the `zone equation' because it must also hold if a face (hkl) of a crystal belongs to a zone [uvw].

Two (non-parallel) nets [(h_1k_1l_1)] and [(h_2k_2l_2)] intersect in a point row with direction [uvw] if the indices satisfy the condition [u:v:w=\left|k_1l_1\atop k_2l_2\right|:\left|l_1h_1\atop l_2h_2\right|:\left|h_1k_1\atop h_2k_2\right|.\eqno (1.1.2.8)]The same condition must be satisfied for a zone axis [uvw] defined by the crystal faces [(h_1k_1l_1)] and [(h_2k_2l_2)].

Three nets [(h_1k_1l_1)], [(h_2k_2l_2)], and [(h_3k_3l_3)] intersect in parallel rows, or three faces with these indices belong to one zone if [\left|\matrix{h_1k_1l_1\cr h_2k_2l_2\cr h_3k_3l_3}\right|=0.\eqno (1.1.2.9)]Two (non-parallel) point rows [[u_1v_1w_1]] and [[u_2v_2w_2]] in the direct lattice are parallel to a family of nets (hkl) if [h:k:l=\left|v_1w_1\atop v_2w_2\right|:\left|w_1u_1\atop w_2u_2\right|:\left|u_1v_1\atop u_2v_2\right|.\eqno (1.1.2.10)]The same condition holds for a face (hkl) belonging to two zones [[u_1v_1w_1]] and [[u_2v_2w_2]].

Three point rows [[u_1v_1w_1]], [[u_2v_2w_2]], and [[u_3v_3w_3]] are parallel to a net (hkl), or three zones of a crystal with these indices have a common face (hkl) if [\left|\matrix{u_1v_1w_1\cr u_2v_2w_2\cr u_3v_3w_3}\right|=0.\eqno (1.1.2.11)]A net (hkl) is perpendicular to a point row [uvw] if [\eqalignno{&{a\over h}(au+bv\cos\gamma+cw\cos\beta)\cr &\quad={b\over k}(au\cos\gamma+bv+cw\cos\alpha)\cr &\quad={c\over l}(au\cos\beta+bv\cos\alpha+cw).& (1.1.2.12)}]








































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