Trigonal and hexagonal 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, pp. 7-9

Section 1.2.5. Trigonal and hexagonal crystal system

E. Koch^a

^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 Mathematical equation 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 Mathematical symbol

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

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 Mathematical symbol and all triplets h, k, l are listed in Table 1.2.5.2. $[{u\over h}+{v+w\over h}\cos\alpha={v\over k}+{u+w\over k}\cos\alpha={w\over l}+{u+v\over l}\cos\alpha, \eqno (1.1.2.12f)]$ $[\eqalignno{ {\bf t}_1\cdot{\bf t}_2 &=(u_1u_2+v_1v_2+w_1w_2)a^2 \cr &\quad +(u_1v_2+u_2v_1+u_1w_2+u_2w_1 \cr &\quad +v_1w_2+v_2w_1)a^2\cos\alpha, &(1.1.3.4f)}]$ $[\eqalignno{ {\bf r}^*_1 \cdot{\bf r}^*_2& =(h_1h_2+k_1k_2+l_1l_2)a^{*2} \cr &\quad +(h_1k_2+h_2k_1+h_1l_2+h_2l_1 \cr &\quad +k_1l_2+k_2l_1)a^{*2}\cos\alpha^*. &(1.1.3.7f)}]$

Table 1.2.5.2| top | pdf |
Assignment of integers $[s_1\le50]$ to triplets h, k, l with Mathematical symbol and to integers

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

s₁	s₂	h	k	l	s₁	s₂	h	k	l	s₁	s₂	h	k	l
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
2	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	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
5	2	2	1	0		−11	4	−3	1	40	−12	−6	2	0
6	−3	−2	1	1		−5	−5	1	0	40	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
10	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	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
13	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	32	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	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
20	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

References

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