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

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. C. ch. 1.2, pp. 8-9

Section 1.2.5.2. Description referred to rhombohedral axes

E. Kocha

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

1.2.5.2. Description referred to rhombohedral axes

| top | pdf |

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[link]. [{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.

s1s2hkls1s2hkls1s2hkl
1010024−12−42238−19−532
2−1−110−44−22−11−611
111020422 5−32
3−1−11125−12−430−16−11
31110500 53−2
402001243013611
5−2−21026−13−43131532
2210−114−3140−12−620
6−3−211−5−51012620
−12−11551041−20−540
5211 43−1−16−621
8−4−22019431 −443
 422027−9−511−86−21
9−4−221 −333 44−3
0300−15−11462−1
 22−11151120621
822127333 540
10−3−31029−14−43240443
3310−10−52042−21−541
11−5−311 4−32−195−41
−13−11−243−21154−1
73111052029541
12−4−2222643243−21−533
1222230−13−521−95−33
13−6−320−75−2139533
6320352−144−20−622
14−7−32117521−46−22
−53−2132−16−44028622
132−11644045−22−542
1132133−16−522−18−630
160400 −441 5−42
17−8−322−45−22254−2
−4−410844−118630
 3−222452238542
4410 44146−21−631
1632234−15−530−156−31
18−9−330 −433963−1
−7−411−94−3327631
−14−111553048−16−444
94113343348444
 33035−17−53149−24−632
19−9−331−135−31−126−32
333−1753−10700
1533123531 63−2
20−8−42036−16−44236632
8420060050−25−550
21−10−421 44−2−23−543
−64−2132442−175−43
242−137−6−610−7−710
144216610 54−3
22−9−332    7710
−333−2    25550
21332    47543








































to end of page
to top of page