Description referred to hexagonal axes

Koch, E.

doi:10.1107/97809553602060000573

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

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International Tables for Crystallography (2006). Vol. C. ch. 1.2, pp. 7-8

Section 1.2.5.1. Description referred to hexagonal axes

E. Koch^a

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

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. $[{2u-v\over 2h}={2v-u\over 2k}={c^2w\over a^2l}, \eqno (1.1.2.12e)]$ $[{\bf t}_1\cdot{\bf t}_2=(u_1u_2+v_1v_2- \textstyle{1\over2}u_1v_2-{1\over2}u_2v_1)a^2+w_1w_2c^2, \eqno (1.1.3.4e)]$ $[{\bf r}^*_1\cdot{\bf r}^*_2=(h_1h_2+k_1k_2+ \textstyle{1\over2}h_1k_2+ {1\over2}h_2k_1)a^{*2}+ l_1l_2c^{*2}. \eqno (1.1.3.7e)]$

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.

s	h	k	s	h	k	s	h	k
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	49	5	3	91	9	1
19	3	2	52	6	2	91	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

References

International Tables for Crystallography (2006). Vol. C. ch. 1.2, pp. 7-8