Tetragonal crystal system

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, p. 7

Section 1.2.4. Tetragonal crystal system

E. Koch^a

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

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

References

International Tables for Crystallography (2006). Vol. C. ch. 1.2, p. 7