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. 9

Section 1.2.6. Cubic crystal system

E. Kocha

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

1.2.6. Cubic crystal system

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

Bravais lattice types: cP, cI, cF

Symmetry of lattice points: [m\bar3m]

Simplified formulae: [V=({\bf abc})=\left[\left| \matrix{ a^2&0&0 \cr0&a^2&0 \cr 0&0&a^2} \right|\right]^{1/2}=a^3, \eqno (1.1.1.1g)] [a^*=b^*=c^*={1\over a}, \quad \alpha^*=\beta^*=\gamma^*=90^\circ, \eqno (1.1.1.3g)] [V^*=({\bf a}^*{\bf b}^*{\bf c}^*)=\left[\left| \matrix{ a^{*2}&0&0 \cr 0&a^{*2}&0 \cr 0&0&a^{*2}}\right|\right]^{1/2}=a^{*3}=a^{-3}, \eqno (1.1.1.4g)] [a=b=c={1\over a^*}, \quad \alpha=\beta=\gamma=90^\circ, \eqno (1.1.1.7g)] [t^2=(u^2+v^2+w^2)a^2, \eqno (1.1.2.1g)] [r^{*2}=(h^2+k^2+l ^2)a^{*2}=sa^{*2} \eqno (1.1.2.2g)]with [s=h^2+k^2+l ^2.]For each value of [s\le100], all corresponding triplets h, k, l are listed in Table 1.2.6.1[link]. [{u\over h}={v\over k}={w\over l}, \eqno (1.1.2.12g)] [{\bf t}_1\cdot {\bf t}_2=(u_1u_2+v_1v_2+w_1w_2)a^2, \eqno (1.1.3.4g)] [{\bf r}^*_1\cdot{\bf r}^*_2=(h_1h_2+k_1k_2+l_1l_2)a^{*2}. \eqno (1.1.3.7g)]

Table 1.2.6.1| top | pdf |
Assignment of integers [s\le 100] to triplets h, k, l with [s=h^2+k^2+l^2]

Each triplet represents all 48 triplets resulting from permutations and sign combinations.

shk lshk lshk lshk lshk lshk l
1 1 0 0 25 5 0 0 42 5 4 1 59 7 3 1 74 8 3 1 88 6 6 4
2 1 1 0 4 3 0 43 5 3 3 5 5 3 7 5 0 89 9 2 2
3 1 1 1 26 5 1 0 44 6 2 2 61 6 5 0 7 4 3 8 5 0
4 2 0 0 4 3 1 45 6 3 0 6 4 3 75 7 5 1 8 4 3
5 2 1 0 27 5 1 1 5 4 2 62 7 3 2 5 5 5 7 6 2
6 2 1 1 3 3 3 46 6 3 1 6 5 1 76 6 6 2 90 9 3 0
8 2 2 0 29 5 2 0 48 4 4 4 64 8 0 0 77 8 3 2 8 5 1
9 3 0 0 4 3 2 49 7 0 0 65 8 1 0 6 5 4 7 5 4
2 2 1 30 5 2 1 6 3 2 7 4 0 78 7 5 2 91 9 3 1
10 3 1 0 32 4 4 0 50 7 1 0 6 5 2 80 8 4 0 93 8 5 2
11 3 1 1 33 5 2 2 5 5 0 66 8 1 1 81 9 0 0 94 9 3 2
12 2 2 2 4 4 1 5 4 3 7 4 1 8 4 1 7 6 3
13 3 2 0 34 5 3 0 51 7 1 1 5 5 4 7 4 4 96 8 4 4
14 3 2 1 4 3 3 5 5 1 67 7 3 3 6 6 3 97 9 4 0
16 4 0 0 35 5 3 1 52 6 4 0 68 8 2 0 82 9 1 0 6 6 5
17 4 1 0 36 6 0 0 53 7 2 0 6 4 4 8 3 3 98 9 4 1
3 2 2 4 4 2 6 4 1 69 8 2 1 83 9 1 1 8 5 3
18 4 1 1 37 6 1 0 54 7 2 1 7 4 2 7 5 3 7 7 0
3 3 0 38 6 1 1 6 3 3 70 6 5 3 84 8 4 2 99 9 3 3
19 3 3 1 5 3 2 5 5 2 72 8 2 2 85 9 2 0 7 7 1
20 4 2 0 40 6 2 0 56 6 4 2 6 6 0 7 6 0 7 5 5
21 4 2 1 41 6 2 1 57 7 2 2 73 8 3 0 86 9 2 1 100 10 0 0
22 3 3 2 5 4 0 5 4 4 6 6 1 7 6 1 8 6 0
24 4 2 2 4 4 3 58 7 3 0       6 5 5      








































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