Table 8.3.1.1

Prince, E.; Finger, L. W.; Konnert, J. H.

doi:10.1107/97809553602060000611

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. 8.3, pp. 695-696

Table 8.3.1.1

E. Prince,^a L. W. Finger^b and J. H. Konnert^c

^a NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA,^bGeophysical Laboratory, Carnegie Institution of Washington, 5251 Broad Branch Road NW, Washington, DC 20015-1305, USA, and ^cLaboratory for the Structure of Matter, Code 6030, Naval Research Laboratory, Washington, DC 20375-5000, USA

Table 8.3.1.1 | top | pdf |
Symmetry conditions for second-cumulant tensors

If more than one condition is applicable for a space group, the site is identified by its Wyckoff notation following the space-group symbol. The stated conditions are valid only for the first equipoint listed for the position. For space groups with alternative choices of origin, the option with a centre of symmetry has been selected.

(A) Monoclinic.

(1) Site symmetry m, 2, 2/m – four independent elements

(a) β₁₂ = β₂₃ = 0; one principal axis parallel to [010]
All groups with unique axis b
(b) β₁₃ = β₂₃ = 0; one principal axis parallel to [001]
All groups with unique axis c

(B) Orthorhombic.

(1) Site symmetry m, 2, 2/m – four independent elements

(a) β₁₂ = β₁₃ = 0; one principal axis parallel to [100]
, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , $[Ibam(\,f)]$ , ,
(b) β₁₂ = β₂₃ = 0; one principal axis parallel to [010]
, , , , $[F222(\,f,i)]$ , , , , , , , , , , , , , , , , $[Pmmm(\,f)]$ , , , , , , , $[Ccca(\,f)]$ , , $[Fddd(\,f)]$ , , , ,
(c) β₁₃ = β₂₃ = 0; one principal axis parallel to [001]
, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

(2) Site symmetry mm2, 222, mmm – three independent elements

(a) β₁₂ = β₁₃ = β₂₃ = 0; principal axes parallel to crystal axes
All space groups

(1) Site symmetry m, 2, 2/m – four independent elements

(a) β₁₂ = β₁₃ = 0; one principal axis parallel to [100]
, $[P4_222(\,j,k,l,m)]$ , , $[I4_122(\,f)]$ , , $[P\overline {4}2m(i,j,k,l)]$ , $[P\overline {4}2c(g,i)]$ , $[I\overline {4}2m(\,f,g)]$ , $[I\overline {4}2d(d)]$ , , , , , , , , , , , $[I4/mcm(\,j)]$ , ,
(b) β₁₂ = β₂₃ = 0; one principal axis parallel to [010]
, , , , , $[P\overline {4}2c(h,j)]$ , $[P\overline {4}m2(\,j,k)]$ , $[I\overline {4}m2(i)]$ ,
(c) β₁₃ = β₂₃ = 0; one principal axis parallel to [001]
, , , , $[P\overline {4}]$ , $[I\overline {4}]$ , , , , , , , , , , , $[I422(\,f)]$ , , , , , , , , $[P\overline {4}2m(m)]$ , $[P\overline {4}2c(k,l,m)]$ , $[P\overline {4}2_1m(d)]$ , $[P\overline {4}21c]$ , $[P\overline {4}c2(g,h,i)]$ , $[P\overline {4}b2(e,f)]$ , $[P\overline {4}n2(e,h)]$ , $[I\overline {4}c2(\,f,g)]$ , $[I\overline {4}2m(h)]$ , $[I\overline {4}2d(c)]$ , $[P4/mmm(\,p,q)]$ , , , , , , , $[P4_2/mcm(\,f,k,n)]$ , $[P4_2/nbc(\,f,g)]$ , , , , $[P4_2/ncm(\,f)]$ , , ,
(d) β₁₁ = β₂₂, β₁₃ = −β₂₃; one principal axis parallel to [110]
$[P422(\,j,k)]$ , , , , , , , , , , $[P\overline {4}m2(h,i)]$ , $[P\overline {4}c2(e,f)]$ , $[P\overline {4}b2(g,h)]$ , $[P\overline {4}n2(g)]$ , $[I\overline {4}m2(g,h)]$ , $[I\overline {4}c2(e,h)]$ , $[P4/mcc(\,j)]$ , , , , , $[P4_2/nbc(\,j)]$ , , , , , , $[I4_1/acd(\,f)]$
(e) β₁₁ = β₂₂, β₁₃ = β₂₃; one principal axis parallel to $[[1\overline {1}0]]$
, , , , , , , $[P\overline {4}2m(n)]$ , $[P\overline {4}2_1m(e)]$ , $[P\overline {4}n2(\,f)]$ , $[I\overline {4}2m(i)]$ , , , , $[P4/ncc(\,f)]$ , , $[P4_2/mnm(\,j)]$ , $[P4_2/nmc(\,f)]$ , , $[I4/mmm(\,f,m)]$ ,

(2) Site symmetry mm2, 222, mmm – three independent elements

(a) β₁₂ = β₁₃ = β₂₃ = 0; principal axes parallel to crystal axes
, , , , , , , $[ P\overline {4}2m(e,f)]$ , $[ P\overline {4}2c]$ , $[ P\overline {4}m2]$ , $[ I\overline {4}m2]$ , $[ I\overline {4}2m(c)]$ , , , , , , , , , , ,
(b) β₁₁ = β₂₂, β₁₃ = β₂₃ = 0; principal axes parallel to [110], $[[1\overline {1}0]]$ and [001]
, , , , , , , , , $[P\overline {4}2m(g,h)]$ , $[P\overline {4}2_1m]$ , $[P\overline {4}c2]$ , $[P\overline {4}b2]$ , $[P\overline {4}n2]$ , $[I\overline {4}c2]$ , $[I\overline {4}2m(e)]$ , $[P4/mmm(\,j,k)]$ , , , , , , , , , , , , ,

(3) Site symmetry 4, $[\overline {4}]$ , 4/m, 4mm, $[\overline {4}2m]$ , 422, 4/mmm – two independent elements

(a) β₁₁ = β₂₂, β₁₂ = β₁₃ = β₂₃ = 0; uniaxial with unique axis parallel to [001]
All space groups

(D) Trigonal (hexagonal axes) and hexagonal.

(1) Site symmetry m, 2, 2/m – four independent elements

(a) β₁₃ = β₂₃ = 0; one principal axis parallel to [001]
, , , $[P\overline 6]$ , , , , , , , $[P \overline 6m2(l,m)]$ , $[P\overline 6c2(k)]$ , $[P\overline 62m(\,j,k)]$ , $[P\overline 62c(h)]$ , $[P6/mmm(\,p,q)]$ , , $[P6_3/mcm(\,j)]$ , $[P6_3/mmc(\,j)]$
(b) β₁₁ = β₂₂, β₁₃ = −β₂₃; one principal axis parallel to [110]
, , $[P\overline 3m1(i)]$ , $[R\overline 3m(h)]$ , , , $[P\overline 6m2(n)]$
(c) β₁₁ = β₂₂, β₁₃ = β₂₃; one principal axis parallel to $[[2\overline 10]]$
, , , $[P\overline 31m(i,j)]$ , $[P\overline 31c]$ , , $[P\overline 6c2(\;j)]$
(d) β₂₂ = 2β₁₂, 2β₁₃ = β₂₃; one principal axis parallel to [100]
, , , , $[P\overline {3}m1(e,f,g,h)]$ , $[P\overline {3}c1]$ , $[R\overline {3}m(d,e,f,g)]$ , $[R\overline {3}c]$ , $[P622(\,j,k)]$ , , , , , , $[P\overline {6}2c(g)]$ , , $[P6/mcc(\,j)]$ ,
(e) β₂₂ = 2β₁₂, β₂₃ = 0; one principal axis parallel to [210]
, $[P\overline {3}1m(\,f,g,k)]$ , , , , , , , , $[P\overline {6}2m(i)]$ , , , $[P6_3/mcm(\,f,i,k)]$

(2) Site symmetry mm2, 222, mmm – three independent elements

(a) β₂₂ = 2β₁₂, β₁₃ = β₂₃ = 0; principal axes parallel to [100] and [001]
, , , , $[P\overline {6}2m]$ , , , ,
(b) β₁₁ = β₂₂, β₁₃ = β₂₃ = 0; principal axes parallel to [110], $[[2\overline {1}0]]$ and [001]
$[P\overline {6}m2]$

(3) Site symmetry 3, $[\overline {3}]$ , 3m, 32, $[\overline {3}m]$ , $[\overline {6}]$ , 6, 6/m, $[\overline {6}m2]$ , 6mm, 622, [6/mmm]

– two independent elements

(a) β₁₁ = β₂₂ = 2β₁₂, β₁₃ = β₂₃ = 0; unique axis parallel to c
All space groups

(E) Cubic.

(1) Site symmetry m, 2, 2/m – four independent elements

(a) β₁₂ = β₁₃ = 0; one principal axis parallel to [100]
, , , , $[Pm\overline {3}]$ , $[Pn\overline {3}]$ , $[Fm\overline {3}]$ , $[Fd\overline {3}]$ , $[Im\overline {3}]$ , $[Ia\overline {3}]$ , , , , $[F4_132(\,f)]$ , , $[I4_132(\,f)]$ , $[P\overline {4}3m(h)]$ , $[I\overline {4}3m(\,f)]$ , $[P\overline {4}3n]$ , $[F\overline {4}3c]$ , $[I\overline {4}3d]$ , $[Pm\overline {3}m(k,l)]$ , $[Pn\overline {3}n(g)]$ , $[Pm\overline {3}n(k)]$ , $[Pn\overline {3}m(h)]$ , $[Fm\overline {3}m(\,j)]$ , $[Fm\overline {3}c(i)]$ , $[Fd\overline {3}c(\,f)]$ , $[Im\overline {3}m(\,j)]$ , $[Ia\overline {3}d(\,f)]$
(b) β₁₁ = β₂₂, β₁₃ = β₂₃; one principal axis parallel to $[[1\overline {1}0]]$
$[P\overline {4}3m(i)]$ , $[F\overline {4}3m]$ , $[I\overline {4}3m(g)]$ , $[Pm\overline {3}m(m)]$ , $[Pn\overline {3}m(k)]$ , $[Fm\overline {3}m(k)]$ , $[Fd\overline {3}m(g)]$ , $[Im\overline {3}m(k)]$
(c) β₂₂ = β₃₃, β₁₂ = −β₁₃; one principal axis parallel to [011]
, , , , , , $[Pn\overline {3}n(h)]$ , $[Pm\overline {3}n(\,j)]$ , $[Pn\overline {3}m(\,j)]$ , $[Fm\overline {3}c(h)]$
(d) β₂₂ = β₃₃, β₁₂ = β₁₃; one principal axis parallel to $[[01\overline 1]]$
, , , , , $[Pn\overline 3m(i)]$ , $[Fd\overline 3m(h,i)]$ , $[Fd\overline 3c(g)]$ , $[Im\overline 3m(i)]$ , $[Ia\overline 3d(g)]$

(2) Site symmetry mm2, 222, mmm – three independent elements

(a) β₁₂ = β₁₃ = β₂₃ = 0; principal axes parallel to crystal axes
, , $[Pm\overline 3]$ , $[Pn\overline 3]$ , $[Fm\overline 3]$ , $[Im\overline 3]$ , , $[P\overline 43n]$ , $[Pm\overline 3m(h)]$ , $[Pm\overline 3n]$ , $[Fm\overline 3c]$ , $[Im\overline 3m(g)]$
(b) β₂₂ = β₃₃, β₁₂ = β₁₃ = 0; principal axes parallel to [011], $[[01\overline 1]]$ and [100]
, , , , $[P\overline 43m]$ , $[F\overline 43m]$ , $[I\overline 43m]$ , $[Pm\overline 3m(i,j)]$ , $[Pn\overline 3m]$ , $[Fm\overline 3m]$ , $[Fd\overline 3m]$ , $[Im\overline 3m(h)]$ , $[Ia\overline 3d]$

(3) Site symmetry 3, $[\overline 3]$ , 3m, 32, $[\overline 3m]$ , $[\overline 6]$ , 6, 6/m, $[\overline 6m2]$ , 6mm, 622, 6/mmm – two independent elements

(a) β₁₁ = β₂₂ = β₃₃, β₁₂ = β₁₃ = β₂₃; unique axis parallel to [111]
All space groups

(4) Site symmetry 4, $[\overline {4}]$ , 4/m, 4mm, $[\overline {4}2m]$ , 422, 4/mmm – two independent elements

(a) β₂₂ = β₃₃, β₁₂ = β₁₃ = β₂₃ = 0; uniaxial with unique axis parallel to [100]
All space groups

(5) Site symmetry 23, m3, $[\overline {4}3m]$ , 432, m3m – one independent element

(a) β₁₁ = β₂₂ = β₃₃, β₁₂ = β₁₃ = β₂₃ = 0; isotropic
All space groups