Monoclinic crystal system

Koch, E.

doi:10.1107/97809553602060000573

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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. 1.2, p. 6

Section 1.2.2. Monoclinic crystal system

E. Koch^a

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

1.2.2. Monoclinic crystal system

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Bravais lattice types: mP, mS

Symmetry of lattice points: 2/m

1.2.2.1. Setting with `unique axis b'

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Metrical conditions: a, b, c, β arbitrary; α = γ = 90°

Bravais lattice types: mP, mC or mA or mI

Symmetry of lattice points: .2/m.

Simplified formulae: $[V=({\bf abc})=\left[\left|\matrix{a^{2}&0&ac\cos \beta \cr 0&b^{2}&0\cr ac\cos \beta&0&c^{2}}\right|\right]^{1/2} =abc\sin\beta, \eqno(1.1.1.1a)]$ $[\eqalign{ &a^*={1\over a\sin \beta},\quad b^*={1\over b},\quad c^*={1\over c\sin \beta}, \cr &\alpha^*=\gamma^*=90^\circ,\quad \beta^*=180^\circ - \beta, } \Biggr\rbrace \eqno(1.1.1.3a)]$ $[\eqalignno{V^* & =({\bf a}^{*}{\bf b}^{*}{\bf c}^{*})=\left[\left|\matrix{ a^{*2}&0&a^*c^*\cos\beta^* \cr 0&b^{*2}&0 \cr a^*c^*\cos\beta^*&0&c^{*2}}\right|\right]^{1/2} \cr &=a^*b^*c^*\sin\beta^*, & (1.1.1.4a)}]$ $[\left. \eqalign{ a&={1\over a^*\sin\beta^*},\quad b={1\over b^*},\quad c={1\over c^*\sin\beta^*}, \cr \alpha&=\gamma=90^\circ,\quad\beta=180^\circ-\beta^*,}\right\} \eqno (1.1.1.7a)]$ $[t^2=u^2a^2+v^2b^2+w^2c^2+2uwac\cos\beta, \eqno (1.1.2.1a)]$ $[r^{*2}=h^2a^{*2}+k^2b^{*2}+l^2c^{*2}+2hla^*c^*\cos\beta^*, \eqno (1.1.2.2a)]$ $[{a\over h}(au+cw\cos\beta)={b^2v\over k}={c\over l}(au\cos\beta+cw), \eqno (1.1.2.12a)]$ $[\eqalignno{ {\bf t}_1\cdot {\bf t}_2 &=u_1u_2a^2+v_1v_2b^2+w_1w_2c^2 \ \, \cr &\quad+(u_1w_2+u_2w_1)ac\cos\beta, & (1.1.3.4a)}]$ $[\eqalignno{ {\bf r}^*_1\cdot {\bf r}^*_2 &= h_1h_2a^{*2}+k_1k_2b^{*2}+l_1l_2c^{*2} \cr &\quad+(h_1l_2+h_2l_1)a^*c^*\cos\beta^*. &(1.1.3.7a)}]$

1.2.2.2. Setting with `unique axis c'

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Metrical conditions: a, b, c, γ arbitrary; α = β = 90°

Bravais lattice types: mP, mB or mA or mI

Symmetry of lattice points: ..2/m

Simplified formulae: $[V=({\bf abc})=\left[\left| \matrix{ a^2&ab\cos\gamma&0 \cr ab\cos\gamma&b^2&0 \cr 0&0&c^2}\right|\right]^{1/2}=abc\sin\gamma, \eqno (1.1.1.1b)]$ $[\eqalign{&a^*={1\over a\sin\gamma}, \quad b^*={1\over b\sin\gamma}, \quad c^*={1\over c}, \cr &\alpha^*=\beta^*=90^\circ, \quad \gamma^*=180^\circ - \gamma, } \Biggr\rbrace \eqno(1.1.1.3b)]$ $[\eqalignno{V^*&=({\bf a}^*{\bf b}^*{\bf c}^*)=\left[\left| \matrix{ a^{*2}&a^*b^*\cos\gamma^*&0 \cr a^*b^*\cos\gamma^*&b^{*2}&0 \cr 0&0&c^{*2}}\right|\right]^{1/2} \cr &=a^*b^*c^*\sin\gamma^*, &(1.1.1.4b)}]$ $[\left. \eqalign{ a&={1\over a^*\sin\gamma^*}, \quad b={1\over b^*\sin\gamma^*}, \quad c={1\over c^*}, \cr \alpha &=\beta=90^\circ, \quad \gamma=180^\circ-\gamma^*,}\right\} \eqno (1.1.1.7b)]$ $[t^2=u^2a^2+v^2b^2+w^2c^2+2uvab\cos\gamma, \eqno (1.1.2.1b)]$ $[r^{*2}=h^2a^{*2}+k^2b^{*2}+l^2c^{*2}+2hka^*b^*\cos\gamma^*,\eqno (1.1.2.2b)]$ $[{a\over h}(au+bv\cos\gamma)={b\over k}(au\cos\gamma+bv)={c^2w\over l}, \eqno (1.1.2.12b)]$ $[\eqalignno{ {\bf t}_1\cdot{\bf t}_2&=u_1u_2a^2+v_1v_2b^2+w_1w_2c^2 \cr &\quad +(u_1v_2+u_2v_1)ab\cos\gamma, &(1.1.3.4b)}]$ $[\eqalignno{ {\bf r}^*_1\cdot{\bf r}^*_2&=h_1h_2a^{*2}+k_1k_2b^{*2}+l_1l_2c^{*2} \cr &\quad +(h_1k_2+h_2k_1)a^*b^*\cos\gamma^*. & (1.1.3.7b)}]$

References

International Tables for Crystallography (2006). Vol. C. ch. 1.2, p. 6