Angles in direct and reciprocal space

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. 4-5

Section 1.1.3. Angles in direct and reciprocal space

E. Koch^a

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

1.1.3. Angles in direct and reciprocal space

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The angles between the normal of a crystal face and the basis vectors a, b, c are called the direction angles of that face. They may be calculated as angles between the corresponding reciprocal-lattice vector r* and the basis vectors $[\lambda={\bf r}^*\wedge{\bf a}]$ , $[\mu={\bf r}^*\wedge{\bf b}]$ and $[\nu={\bf r}^*\wedge{\bf c}]$ : $[\left.\matrix{\cos\lambda=\displaystyle{h\over a}d(hkl),\quad\cos\mu={k\over b}d(hkl),\cr \noalign{\vskip5.5pt}\cos\nu=\displaystyle{l\over c}d(hkl).}\right\}\eqno (1.1.3.1)]$ The three equations can be combined to give $[ \left.\matrix{&\quad&\displaystyle a:b:c={h\over\cos\lambda}:{k\over\cos\mu}:{l\over\cos\nu}\cr {\hskip-32pt{\rm or}}\cr &&h:k:l=a\cos\lambda:b\cos\mu: c\cos\nu.}\right\}\eqno (1.1.3.2)]$ The first formula gives the ratios between a, b, and c, if for any face of the crystal the indices (hkl) and the direction angles λ, μ, and ν are known. Once the axial ratios are known, the indices of any other face can be obtained from its direction angles by using the second formula.

Similarly, the angles between a direct-lattice vector t and the reciprocal basis vectors $[\lambda^*={\bf t}\wedge{\bf a}^*]$ , $[\mu^*={\bf t}\wedge{\bf b}^*]$ and $[\nu^*={\bf t}\wedge{\bf c}^*]$ are given by $[ \left.\matrix{\displaystyle\cos\lambda^*={u\over a^*}d^*(uvw),\quad\cos\mu^*={v\over b^*}d^*(uvw),\cr\noalign{\vskip5.5pt}\displaystyle\cos\nu^*={w\over c^*}d^*(uvw).}\right\}\eqno (1.1.3.3)]$ The angle $[\psi]$ between two direct-lattice vectors $[{\bf t}_1]$ and $[{\bf t}_2]$ or between two corresponding point rows [[u_1v_1w_1]] and [[u_2v_2w_2]] may be derived from the scalar product $[\eqalignno{{\bf t}_1\cdot{\bf t}_2&=u_1u_2a^2+v_1v_2b^2+w_1w_2c^2+(u_1v_2+u_2v_1)ab\cos\gamma\cr &\quad+(u_1w_2+u_2w_1)ac\cos\beta+(v_1w_2+v_2w_1)bc\cos\alpha\cr&& (1.1.3.4)}]$ as $[\cos\psi={{\bf t}_1\cdot{\bf t}_2\over t_1t_2}.\eqno (1.1.3.5)]$ Analogously, the angle $[\varphi]$ between two reciprocal-lattice vectors $[{\bf r}^*_1]$ and $[{\bf r}^*_2]$ or between two corresponding point rows [[h_1k_1l_1]^*] and [[h_2k_2l_2]^*] or between the normals of two corresponding crystal faces [(h_1k_1l_1)] and [(h_2k_2l_2)] may be calculated as $[\cos\varphi={{\bf r}^*_1\cdot{\bf r}^*_2\over r_1r_2}\eqno (1.1.3.6)]$ with $[\eqalignno{{\bf r}^*_1\cdot{\bf r}^*_2&=h_1h_2a^{*2}+k_1k_2b^{*2}+l_1l_2c^{*2}\cr &\quad+(h_1k_2+h_2k_1)a^*b^*\cos\gamma^*\cr &\quad+(h_1l_2+h_2l_1)a^*c^*\cos\beta^*\cr &\quad+(k_1l_2+k_2l_1)b^*c^*\cos\alpha^*. & (1.1.3.7)}]$

Finally, the angle $[\omega]$ between a first direction [uvw] of the direct lattice and a second direction [hkl] of the reciprocal lattice may also be derived from the scalar product of the corresponding vectors t and r*. $[\cos\omega={{\bf t}\cdot{\bf r}^*\over tr^*}={uh+vk+wl\over tr^*}.\eqno (1.1.3.8)]$

References

International Tables for Crystallography (2006). Vol. C. ch. 1.1, pp. 4-5