Cubic 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. 9

Section 1.2.6. Cubic crystal system

E. Koch^a

^a Institut 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 Mathematical equation For each value of $[s\le100]$ , all corresponding triplets h, k, l are listed in Table 1.2.6.1. $[{u\over h}={v\over k}={w\over l}, \eqno (1.1.2.12g)]$ $[{\bf t}_1\cdot {\bf t}_2=(u_1u_2+v_1v_2+w_1w_2)a^2, \eqno (1.1.3.4g)]$ $[{\bf r}^*_1\cdot{\bf r}^*_2=(h_1h_2+k_1k_2+l_1l_2)a^{*2}. \eqno (1.1.3.7g)]$

Table 1.2.6.1| top | pdf |
Assignment of integers $[s\le 100]$ to triplets h, k, l with Mathematical symbol

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

s	h	k	l	s	h	k	l	s	h	k	l	s	h	k	l	s	h	k	l	s	h	k	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	25	4	3	0	43	5	3	3	59	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	26	4	3	1	45	6	3	0	61	6	4	3	75	7	5	1		8	4	3
5	2	1	0	27	5	1	1	45	5	4	2	62	7	3	2	75	5	5	5		7	6	2
6	2	1	1	27	3	3	3	46	6	3	1	62	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	29	4	3	2	49	7	0	0	65	8	1	0	77	6	5	4		7	5	4
9	2	2	1	30	5	2	1	49	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	33	4	4	1		5	4	3		7	4	1		8	4	1	94	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	34	4	3	3	51	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	97	6	6	5
17	4	1	0	36	6	0	0	53	7	2	0	68	6	4	4	82	8	3	3	98	9	4	1
17	3	2	2	36	4	4	2	53	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	69	7	4	2	83	7	5	3		7	7	0
18	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	38	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	72	6	6	0	85	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	57	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

References

International Tables for Crystallography (2006). Vol. C. ch. 1.2, p. 9