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

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. C. ch. 1.2, pp. 7-9

Section 1.2.5. Trigonal and hexagonal crystal system

E. Kocha

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

1.2.5. Trigonal and hexagonal crystal system

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1.2.5.1. Description referred to hexagonal axes

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Metrical conditions: a = b; c arbitrary; α = β = 90°; γ = 120°

Bravais lattice types: hP, hR

Symmetry of lattice points: 6/mmm (hP), [\bar3m ] (hR)

Simplified formulae: [V=({\bf abc}) =\left [\left| \matrix{a^{2} &-{1\over2} a^{2} &0 \cr -{1\over2}a^{2} &a^2 &0 \cr 0&0&c^2} \right| \right]^{1/2} =\textstyle{1\over2}{\sqrt3} \,\, a^2c, \eqno (1.1.1.1e)] [\left. \eqalign{ a^*&=b^*={\textstyle{2\over3}}\sqrt3 {1\over a}, \quad c^*={1\over c}, \cr \alpha^*&=\beta^*=90^\circ, \quad \gamma^*=60^\circ,} \right\} \eqno (1.1.1.3e)] [\eqalignno{\qquad V^*& =({\bf a}^*{\bf b}^*{\bf c}^*)=\left[\left| \matrix{ a^{*2}&{1\over2}a^{*2}&0 \cr {1\over2}a^{*2}&a^{*2}&0 \cr 0&0&c^{*2}}\right|\right]^{1/2} \cr &=\textstyle{1\over2}\sqrt3\; a^{*2}c^*={2\over3}\sqrt3\; a^{-2}c^{-1}, &(1.1.1.4e)}] [a=b={\textstyle{2\over3}}\sqrt3{1\over a^*}, \quad c={1\over c^*}, \quad \alpha=\beta=90^\circ, \quad \gamma=120^\circ, \eqno (1.1.1.7e)] [t^2=(u^2+v^2-uv)a^2+w^2c^2, \eqno (1.1.2.1e)] [r^{*2}=(h^2+k^2+hk)a^{*2}+l ^2c^{*2}=sa^{*2}+l^2c^{*2} \eqno (1.1.2.2e)]with [s=h^2+k^2+hk.]For each value of [s\le100], all corresponding pairs h, k are listed in Table 1.2.5.1[link]. [{2u-v\over 2h}={2v-u\over 2k}={c^2w\over a^2l}, \eqno (1.1.2.12e)] [{\bf t}_1\cdot{\bf t}_2=(u_1u_2+v_1v_2- \textstyle{1\over2}u_1v_2-{1\over2}u_2v_1)a^2+w_1w_2c^2, \eqno (1.1.3.4e)] [{\bf r}^*_1\cdot{\bf r}^*_2=(h_1h_2+k_1k_2+ \textstyle{1\over2}h_1k_2+ {1\over2}h_2k_1)a^{*2}+ l_1l_2c^{*2}. \eqno (1.1.3.7e)]

Table 1.2.5.1| top | pdf |
Assignment of integers [s\le100] to pairs h, k with [s=h^2+k^2+hk]

Each pair h, k represents in addition the pairs k, −h − k and −h − k, h, the permutations of these three, and the six corresponding centrosymmetrical pairs.

shkshkshk
11031516772
31136607381
42037437555
72139527664
93043617973
122248448190
133149708482
1640539191
1932526265
214157719374
255061549783
27336363100100
28426480 

1.2.5.2. Description referred to rhombohedral axes

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Metrical conditions: a = b = c; α = β = γ

Bravais lattice type: hR

Symmetry of lattice points: [\bar3m]

Simplified formulae: [\eqalignno{\qquad\quad V&=({\bf abc})= \left[\left| \matrix{ a^2&a^2\cos\alpha&a^2\cos\alpha \cr a^2\cos\alpha&a^2&a^2\cos\alpha \cr a^2\cos\alpha&a^2\cos\alpha&a^2}\right|\right]^{1/2} \cr &=a^3[1 - 3\cos^2\alpha+2\cos^3\alpha]^{1/2} \cr &=2a^3\bigg[\sin {\textstyle{3\over2}}\, \alpha\sin^3 {\alpha\over2}\, \bigg]^{1/2}, & (1.1.1.1f)}] [\left. \eqalign{ &\cos{\alpha^*\over2}=\cos{\beta^*\over2}=\cos{\gamma^*\over2}={1\over2\cos\alpha/2}, \cr &a^*=b^*=c^*={1\over a\sin\alpha\sin\alpha^*},}\right\} \eqno (1.1.1.3f)] [\eqalignno{\qquad\quad V^*&=({\bf a}^*{\bf b}^*{\bf c}^*) \cr &=\left[\left| \matrix{ a^{*2}&a^{*2}\cos\alpha^*&a^{*2}\cos\alpha^* \cr a^{*2}\cos\alpha^*&a^{*2}&a^{*2}\cos\alpha^* \cr a^{*2}\cos\alpha^*&a^{*2}\cos\alpha^*&a^{*2}}\right|\right]^{1/2} \cr &=a^{*3}[1 - 3\cos^2\alpha^*+2\cos^3\alpha^*]^{1/2} \cr &=2a^{*3}\bigg[\sin \textstyle{3\over2}\alpha^*\sin^3\displaystyle{\alpha^*\over2}\bigg]^{1/2}, & (1.1.1.4f)}] [\left. \eqalign{ &\cos{\alpha\over2}= \cos{\beta\over2}= \cos{\gamma\over2}= {1\over2\cos\alpha^*/2}, \cr &a=b=c={1\over a^*\sin\alpha^*\sin\alpha},}\right\} \eqno (1.1.1.7f)] [t^2=(u^2+v^2+w^2)a^2+2(uv+uw+vw)a^2\cos\alpha, \eqno (1.1.2.1f)] [\eqalignno{\quad\qquad r^{*2}&=(h^2+k^2+l^2)a^{*2}+2(hk+hl+kl)a^{*2}\cos\alpha^* \cr &=s_1a^{*2}+2s_2a^{*2}\cos\alpha^* & (1.1.2.2f)}]with [s_1=h^2+k^2+l ^2 \quad {\rm and} \quad s_2=hk+hl+kl.]For each value of [s_1\le50], all corresponding values of [s_2] and all triplets h, k, l are listed in Table 1.2.5.2[link]. [{u\over h}+{v+w\over h}\cos\alpha={v\over k}+{u+w\over k}\cos\alpha={w\over l}+{u+v\over l}\cos\alpha, \eqno (1.1.2.12f)] [\eqalignno{ {\bf t}_1\cdot{\bf t}_2 &=(u_1u_2+v_1v_2+w_1w_2)a^2 \cr &\quad +(u_1v_2+u_2v_1+u_1w_2+u_2w_1 \cr &\quad +v_1w_2+v_2w_1)a^2\cos\alpha, &(1.1.3.4f)}] [\eqalignno{ {\bf r}^*_1 \cdot{\bf r}^*_2& =(h_1h_2+k_1k_2+l_1l_2)a^{*2} \cr &\quad +(h_1k_2+h_2k_1+h_1l_2+h_2l_1 \cr &\quad +k_1l_2+k_2l_1)a^{*2}\cos\alpha^*. &(1.1.3.7f)}]

Table 1.2.5.2| top | pdf |
Assignment of integers [s_1\le50] to triplets h, k, l with [s_1 = h^2 +k^2 =l^2] and to integers [s_2=hk+hl+kl]

Each triplet h, k, l represents all twelve triplets resulting from permutation and/or simultaneous change of all signs.

s1s2hkls1s2hkls1s2hkl
1010024−12−42238−19−532
2−1−110−44−22−11−611
111020422 5−32
3−1−11125−12−430−16−11
31110500 53−2
402001243013611
5−2−21026−13−43131532
2210−114−3140−12−620
6−3−211−5−51012620
−12−11551041−20−540
5211 43−1−16−621
8−4−22019431 −443
 422027−9−511−86−21
9−4−221 −333 44−3
0300−15−11462−1
 22−11151120621
822127333 540
10−3−31029−14−43240443
3310−10−52042−21−541
11−5−311 4−32−195−41
−13−11−243−21154−1
73111052029541
12−4−2222643243−21−533
1222230−13−521−95−33
13−6−320−75−2139533
6320352−144−20−622
14−7−32117521−46−22
−53−2132−16−44028622
132−11644045−22−542
1132133−16−522−18−630
160400 −441 5−42
17−8−322−45−22254−2
−4−410844−118630
 3−222452238542
4410 44146−21−631
1632234−15−530−156−31
18−9−330 −433963−1
−7−411−94−3327631
−14−111553048−16−444
94113343348444
 33035−17−53149−24−632
19−9−331−135−31−126−32
333−1753−10700
1533123531 63−2
20−8−42036−16−44236632
8420060050−25−550
21−10−421 44−2−23−543
−64−2132442−175−43
242−137−6−610−7−710
144216610 54−3
22−9−332    7710
−333−2    25550
21332    47543








































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