Lattice vectors, point rows, and net planes

Koch, E.

doi:10.1107/97809553602060000572

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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. Koch^a

^a Institut 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 Mathematical symbol 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 Mathematical symbol .

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 Mathematical symbol . The distance 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 of the reciprocal lattice.

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

Three nets Mathematical symbol , , and 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 and 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 Mathematical symbol and .

Three point rows Mathematical symbol , , and 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 $[\eqalignno{&{a\over h}(au+bv\cos\gamma+cw\cos\beta)\cr &\quad={b\over k}(au\cos\gamma+bv+cw\cos\alpha)\cr &\quad={c\over l}(au\cos\beta+bv\cos\alpha+cw).& (1.1.2.12)}]$

References

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

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

1.1.2. Lattice vectors, point rows, and net planes

References

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

1.1.2. Lattice vectors , point rows, and net planes