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. 6-9
https://doi.org/10.1107/97809553602060000573

## Chapter 1.2. Application to the crystal systems

E. Kocha

aInstitut 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.

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:

### 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:

#### 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:

### 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:

### 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: with For each value of , all corresponding pairs h, k are listed in Table 1.2.4.1.

 Table 1.2.4.1| top | pdf | Assignment of integers to pairs h, k with
 Each pair h, k represents all eight pairs which result from permutation and different sign combinations.
shkshkshk
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 7 6
17 4 1 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
4 3 61 6 5 100 10 0
26 5 1 64 8 0 8 6
29 5 2 65 8 1
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), (hR)

Simplified formulae: with For each value of , all corresponding pairs h, k are listed in Table 1.2.5.1.

 Table 1.2.5.1| top | pdf | Assignment of integers to pairs h, k with
 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.
shkshkshk
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 5 3 91 9 1
19 3 2 52 6 2 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:

Simplified formulae: with For each value of , all corresponding values of and all triplets h, k, l are listed in Table 1.2.5.2.

 Table 1.2.5.2| top | pdf | Assignment of integers to triplets h, k, l with and to integers
 Each triplet h, k, l represents all twelve triplets resulting from permutation and/or simultaneous change of all signs.
s1s2hkls1s2hkls1s2hkl
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
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 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
2 2 1 0 −11 4 −3 1 40 −12 −6 2 0
6 −3 −2 1 1 −5 −5 1 0 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
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 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
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 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 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
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:

Simplified formulae: with For each value of , all corresponding triplets h, k, l are listed in Table 1.2.6.1.

 Table 1.2.6.1| top | pdf | Assignment of integers to triplets h, k, l with
 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

### References

International Tables for Crystallography (2005). Vol. A, Space-group symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer.