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, pp. 7-9

Section 1.2.5. Trigonal and hexagonal crystal system

E. Kocha

aInstitut 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, −hk and −hk, h, the permutations of these three, and the six corresponding centrosymmetrical pairs.

shkshkshk
1 1 0 31 5 1 67 7 2
3 1 1 36 6 0 73 8 1
4 2 0 37 4 3 75 5 5
7 2 1 39 5 2 76 6 4
9 3 0 43 6 1 79 7 3
12 2 2 48 4 4 81 9 0
13 3 1 49 7 0 84 8 2
16 4 0 5 3 91 9 1
19 3 2 52 6 2 6 5
21 4 1 57 7 1 93 7 4
25 5 0 61 5 4 97 8 3
27 3 3 63 6 3 100 10 0
28 4 2 64 8 0  

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
1 0 1 0 0 24 −12 −4 2 2 38 −19 −5 3 2
2 −1 −1 1 0 −4 4 −2 2 −11 −6 1 1
1 1 1 0 20 4 2 2   5 −3 2
3 −1 −1 1 1 25 −12 −4 3 0 −1 6 −1 1
3 1 1 1 0 5 0 0   5 3 −2
4 0 2 0 0 12 4 3 0 13 6 1 1
5 −2 −2 1 0 26 −13 −4 3 1 31 5 3 2
2 2 1 0 −11 4 −3 1 40 −12 −6 2 0
6 −3 −2 1 1 −5 −5 1 0 12 6 2 0
−1 2 −1 1 5 5 1 0 41 −20 −5 4 0
5 2 1 1   4 3 −1 −16 −6 2 1
8 −4 −2 2 0 19 4 3 1   −4 4 3
  4 2 2 0 27 −9 −5 1 1 −8 6 −2 1
9 −4 −2 2 1   −3 3 3   4 4 −3
0 3 0 0 −1 5 −1 1 4 6 2 −1
  2 2 −1 11 5 1 1 20 6 2 1
8 2 2 1 27 3 3 3   5 4 0
10 −3 −3 1 0 29 −14 −4 3 2 40 4 4 3
3 3 1 0 −10 −5 2 0 42 −21 −5 4 1
11 −5 −3 1 1   4 −3 2 −19 5 −4 1
−1 3 −1 1 −2 4 3 −2 11 5 4 −1
7 3 1 1 10 5 2 0 29 5 4 1
12 −4 −2 2 2 26 4 3 2 43 −21 −5 3 3
12 2 2 2 30 −13 −5 2 1 −9 5 −3 3
13 −6 −3 2 0 −7 5 −2 1 39 5 3 3
6 3 2 0 3 5 2 −1 44 −20 −6 2 2
14 −7 −3 2 1 17 5 2 1 −4 6 −2 2
−5 3 −2 1 32 −16 −4 4 0 28 6 2 2
1 3 2 −1 16 4 4 0 45 −22 −5 4 2
11 3 2 1 33 −16 −5 2 2 −18 −6 3 0
16 0 4 0 0   −4 4 1   5 −4 2
17 −8 −3 2 2 −4 5 −2 2 2 5 4 −2
−4 −4 1 0 8 4 4 −1 18 6 3 0
  3 −2 2 24 5 2 2 38 5 4 2
4 4 1 0   4 4 1 46 −21 −6 3 1
16 3 2 2 34 −15 −5 3 0 −15 6 −3 1
18 −9 −3 3 0   −4 3 3 9 6 3 −1
−7 −4 1 1 −9 4 −3 3 27 6 3 1
−1 4 −1 1 15 5 3 0 48 −16 −4 4 4
9 4 1 1 33 4 3 3 48 4 4 4
  3 3 0 35 −17 −5 3 1 49 −24 −6 3 2
19 −9 −3 3 1 −13 5 −3 1 −12 6 −3 2
3 3 3 −1 7 5 3 −1 0 7 0 0
15 3 3 1 23 5 3 1   6 3 −2
20 −8 −4 2 0 36 −16 −4 4 2 36 6 3 2
8 4 2 0 0 6 0 0 50 −25 −5 5 0
21 −10 −4 2 1   4 4 −2 −23 −5 4 3
−6 4 −2 1 32 4 4 2 −17 5 −4 3
2 4 2 −1 37 −6 −6 1 0 −7 −7 1 0
14 4 2 1 6 6 1 0   5 4 −3
22 −9 −3 3 2         7 7 1 0
−3 3 3 −2         25 5 5 0
21 3 3 2         47 5 4 3








































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