Application to the crystal systems

Koch, E.

doi:10.1107/97809553602060000573

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. C. ch. 1.2, pp. 6-9
https://doi.org/10.1107/97809553602060000573

Chapter 1.2. Application to the crystal systems

E. Koch^a

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

In this chapter, all general formulae from Chapter 1.1 are simplified according to the metrical specializations in the monoclinic, orthorhombic, trigonal and hexagonal, tetragonal and cubic crystal systems.

Keywords: crystal systems; crystals; cubic crystal system; hexagonal crystal system; monoclinic crystal system; orthorhombic crystal system; rhombohedral crystal system; tetragonal crystal system; triclinic crystal system; trigonal crystal system.

Information on the description and classification of Bravais lattices, their assignment to crystal systems, the choice of basis vectors for reduced or conventional basis systems, and on basis transformations is given in IT A (2005 , Parts 5 and 9 ). In the following, for each crystal system , the metrical conditions for conventionally chosen basis systems and the possible Bravais types of lattices are listed. As some of the general formulae from Chapter 1.1 become simpler when not applied to a lattice with general (triclinic) metric, these simplified formulae are tabulated for all crystal systems (except triclinic).

Except for triclinic, monoclinic, and orthorhombic symmetry, tables are given that relate pairs h, k or triplets h, k, l of indices to certain sums s of products of these indices needed in equation (1.1.2.2 ). Such tables may be useful, for example, for indexing powder diffraction patterns.

1.2.1. Triclinic crystal system

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No metrical conditions: a, b, c, α, β, γ arbitrary

Bravais lattice type: aP

Symmetry of lattice points: $[\bar 1]$

1.2.2. Monoclinic crystal system

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Bravais lattice types: mP, mS

Symmetry of lattice points: 2/m

1.2.2.1. Setting with `unique axis b'

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

Bravais lattice types: mP, mC or mA or mI

Symmetry of lattice points: .2/m.

Simplified formulae: $[V=({\bf abc})=\left[\left|\matrix{a^{2}&0&ac\cos \beta \cr 0&b^{2}&0\cr ac\cos \beta&0&c^{2}}\right|\right]^{1/2} =abc\sin\beta, \eqno(1.1.1.1a)]$ $[\eqalign{ &a^*={1\over a\sin \beta},\quad b^*={1\over b},\quad c^*={1\over c\sin \beta}, \cr &\alpha^*=\gamma^*=90^\circ,\quad \beta^*=180^\circ - \beta, } \Biggr\rbrace \eqno(1.1.1.3a)]$ $[\eqalignno{V^* & =({\bf a}^{*}{\bf b}^{*}{\bf c}^{*})=\left[\left|\matrix{ a^{*2}&0&a^*c^*\cos\beta^* \cr 0&b^{*2}&0 \cr a^*c^*\cos\beta^*&0&c^{*2}}\right|\right]^{1/2} \cr &=a^*b^*c^*\sin\beta^*, & (1.1.1.4a)}]$ $[\left. \eqalign{ a&={1\over a^*\sin\beta^*},\quad b={1\over b^*},\quad c={1\over c^*\sin\beta^*}, \cr \alpha&=\gamma=90^\circ,\quad\beta=180^\circ-\beta^*,}\right\} \eqno (1.1.1.7a)]$ $[t^2=u^2a^2+v^2b^2+w^2c^2+2uwac\cos\beta, \eqno (1.1.2.1a)]$ $[r^{*2}=h^2a^{*2}+k^2b^{*2}+l^2c^{*2}+2hla^*c^*\cos\beta^*, \eqno (1.1.2.2a)]$ $[{a\over h}(au+cw\cos\beta)={b^2v\over k}={c\over l}(au\cos\beta+cw), \eqno (1.1.2.12a)]$ $[\eqalignno{ {\bf t}_1\cdot {\bf t}_2 &=u_1u_2a^2+v_1v_2b^2+w_1w_2c^2 \ \, \cr &\quad+(u_1w_2+u_2w_1)ac\cos\beta, & (1.1.3.4a)}]$ $[\eqalignno{ {\bf r}^*_1\cdot {\bf r}^*_2 &= h_1h_2a^{*2}+k_1k_2b^{*2}+l_1l_2c^{*2} \cr &\quad+(h_1l_2+h_2l_1)a^*c^*\cos\beta^*. &(1.1.3.7a)}]$

1.2.2.2. Setting with `unique axis c'

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

Bravais lattice types: mP, mB or mA or mI

Symmetry of lattice points: ..2/m

Simplified formulae: $[V=({\bf abc})=\left[\left| \matrix{ a^2&ab\cos\gamma&0 \cr ab\cos\gamma&b^2&0 \cr 0&0&c^2}\right|\right]^{1/2}=abc\sin\gamma, \eqno (1.1.1.1b)]$ $[\eqalign{&a^*={1\over a\sin\gamma}, \quad b^*={1\over b\sin\gamma}, \quad c^*={1\over c}, \cr &\alpha^*=\beta^*=90^\circ, \quad \gamma^*=180^\circ - \gamma, } \Biggr\rbrace \eqno(1.1.1.3b)]$ $[\eqalignno{V^*&=({\bf a}^*{\bf b}^*{\bf c}^*)=\left[\left| \matrix{ a^{*2}&a^*b^*\cos\gamma^*&0 \cr a^*b^*\cos\gamma^*&b^{*2}&0 \cr 0&0&c^{*2}}\right|\right]^{1/2} \cr &=a^*b^*c^*\sin\gamma^*, &(1.1.1.4b)}]$ $[\left. \eqalign{ a&={1\over a^*\sin\gamma^*}, \quad b={1\over b^*\sin\gamma^*}, \quad c={1\over c^*}, \cr \alpha &=\beta=90^\circ, \quad \gamma=180^\circ-\gamma^*,}\right\} \eqno (1.1.1.7b)]$ $[t^2=u^2a^2+v^2b^2+w^2c^2+2uvab\cos\gamma, \eqno (1.1.2.1b)]$ $[r^{*2}=h^2a^{*2}+k^2b^{*2}+l^2c^{*2}+2hka^*b^*\cos\gamma^*,\eqno (1.1.2.2b)]$ $[{a\over h}(au+bv\cos\gamma)={b\over k}(au\cos\gamma+bv)={c^2w\over l}, \eqno (1.1.2.12b)]$ $[\eqalignno{ {\bf t}_1\cdot{\bf t}_2&=u_1u_2a^2+v_1v_2b^2+w_1w_2c^2 \cr &\quad +(u_1v_2+u_2v_1)ab\cos\gamma, &(1.1.3.4b)}]$ $[\eqalignno{ {\bf r}^*_1\cdot{\bf r}^*_2&=h_1h_2a^{*2}+k_1k_2b^{*2}+l_1l_2c^{*2} \cr &\quad +(h_1k_2+h_2k_1)a^*b^*\cos\gamma^*. & (1.1.3.7b)}]$

1.2.3. Orthorhombic crystal system

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

Bravais lattice types: oP, oS (oC, oA), oI, oF

Symmetry of lattice points: mmm

Simplified formulae: $[V=({\bf abc})=\left[\left| \matrix{ a^2&0&0 \cr 0&b^2&0 \cr 0&0&c^2}\right|\right]^{1/2}=abc, \eqno (1.1.1.1c)]$ $[a^*={1\over a}, \quad b^*={1\over b}, \quad c^*={1\over c},\quad \alpha^*=\beta^*=\gamma^*=90^\circ, \eqno (1.1.1.3c)]$ $[\eqalignno{ V^*&=({\bf a}^*{\bf b}^*{\bf c}^*)=\left [\left| \matrix{ a^{*2}&0&0 \cr 0&b^{*2}&0 \cr 0&0&c^{*2}}\right |\right] ^{1/2} \cr &=a^*b^*c^*=a^{-1}b^{-1}c^{-1}, &(1.1.1.4c)}]$ $[a={1\over a^*}, \quad b={1\over b^*}, \quad c={1\over c^*}, \quad\alpha=\beta=\gamma=90^\circ, \eqno (1.1.1.7c)]$ $[t^2=u^2a^2+v^2b^2+w^2c^2, \eqno (1.1.2.1c)]$ $[r^{*2}=h^2a^{*2}+k^2b^{*2}+l^2w^{*2}, \eqno (1.1.2.2c)]$ $[{a^2u\over h}={b^2v\over k}={c^2w\over l}, \eqno (1.1.2.12c)]$ $[{\bf t}_1\cdot{\bf t}_2=u_1u_2a^2+v_1v_2b^2+w_1w_2c^2, \eqno (1.1.3.4c)]$ $[{\bf r}^*_1\cdot {\bf r}^*_2=h_1h_2a^{*2}+k_1k_2b^{*2}+l_1l_2c^{*2}. \eqno (1.1.3.7c)]$

1.2.4. Tetragonal crystal system

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

Bravais lattice types: tP, tI

Symmetry of lattice points: 4/mmm

Simplified formulae: $[V=({\bf abc})=\left[\left| \matrix{ a^2&0&0 \cr 0&a^2&0 \cr 0&0&c^2}\right|\right]^{1/2}=a^2c, \eqno(1.1.1.1d)]$ $[a^*=b^*={1\over a}, \quad c^*={1\over c}, \quad \alpha^*=\beta^*=\gamma^*=90^\circ, \eqno (1.1.1.3d)]$ $[\eqalignno{\qquad\qquad V^*& =({\bf a}^*{\bf b}^*{\bf c}^*)=\left[\left| \matrix{ a^{*2}&0&0 \cr 0&a^{*2}&0 \cr 0&0&c^{*2}}\right|\right]^{1/2} \cr &=a^{*2}c^*=a^{-2}c^{-1}, & (1.1.1.4d)}]$ $[a=b={1\over a^*}, \quad c={1\over c^*}, \quad \alpha=\beta=\gamma=90^\circ, \eqno (1.1.1.7d)]$ $[t^2=(u^2+v^2)a^2+w^2c^2, \eqno (1.1.2.1d)]$ $[r^{*2}=(h^2+k^2)a^{*2}+l ^2c^{*2}=sa^{*2}+l ^2c^{*2} \eqno (1.1.2.2d)]$ with [s=h^2+k^2.] For each value of $[s\le100]$ , all corresponding pairs h, k are listed in Table 1.2.4.1 . $[{u\over h}={v\over k}={c^2w \over a^2l}, \eqno (1.1.2.12d)]$ $[{\bf t}_1\cdot {\bf t}_2 = (u_1u_2+v_1v_2)a^2+w_1w_2c^2, \eqno (1.1.3.4d)]$ $[{\bf r}^*_1\cdot{\bf r}^*_2 = (h_1h_2+k_1k_2)a^{*2}+l_1l_2c^{*2}. \eqno (1.1.3.7d)]$

Table 1.2.4.1| top | pdf |
Assignment of integers $[s\le 100]$ to pairs h, k with [s=h^2+k^2]

Each pair h, k represents all eight pairs which result from permutation and different sign combinations.

s	h	k	s	h	k	s	h	k
1	1	0	32	4	4	68	8	2
2	1	1	34	5	3	72	6	6
4	2	0	36	6	0	73	8	3
5	2	1	37	6	1	74	7	5
8	2	2	40	6	2	80	8	4
9	3	0	41	5	4	81	9	0
10	3	1	45	6	3	82	9	1
13	3	2	49	7	0	85	9	2
16	4	0	50	7	1	85	7	6
17	4	1	50	5	5	89	8	5
18	3	3	52	6	4	90	9	3
20	4	2	53	7	2	97	9	4
25	5	0	58	7	3	98	7	7
25	4	3	61	6	5	100	10	0
26	5	1	64	8	0	100	8	6
29	5	2	65	8	1
			65	7	4

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

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

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

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.

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

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 . $[{u\over h}={v\over k}={w\over l}, \eqno (1.1.2.12g)]$ $[{\bf t}_1\cdot {\bf t}_2=(u_1u_2+v_1v_2+w_1w_2)a^2, \eqno (1.1.3.4g)]$ $[{\bf r}^*_1\cdot{\bf r}^*_2=(h_1h_2+k_1k_2+l_1l_2)a^{*2}. \eqno (1.1.3.7g)]$

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.

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 (2005). Vol. A, Space-group symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 1.2, pp. 6-9
https://doi.org/10.1107/97809553602060000573