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

International Tables for Crystallography (2006). Vol. C, ch. 8.3, pp. 695-696

Table 8.3.1.1 

E. Prince,a L. W. Fingerb and J. H. Konnertc

aNIST 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]

    [P222(i,j,k,l)], [P222_1(a,b)], [C222_1(a)], [C222(e,f)], [F222(e,j)], [I222(e,f)], [I2_12_12_1(a)], [Pmm2(g,h)], [Pmc2_1], [Pma2(c)], [Pmn2_1], [Cmm2(e)], [Cmc2_1], [Amm2(d,e)], [Ama2(b)], [Fmm2(c)], [Imm2(d)], [Ima2(b)], [Pmmm(u,v)], [Pnnn(g,h)], [Pccm(i,j)], [Pban(g,h)], [Pmma(k)], [Pnna(d)], [Pmna(a,b,c,d,e,f,h)], [Pbcm(c)], [Pmmn(e)], [Cmcm(a,b,e,f)], [Cmca(a,b,d,f)], [Cmmm(n)], [Cccm(g)], [Cmma(c,d,h,i,m)], [Ccca(e)], [Fmmm(c,l,m)], [Fddd(e)], [Immm(l)], [Ibam(\,f)], [Ibca(c)], [Imma(a,b,f,h)]

  • (b) β12 = β23 = 0; one principal axis parallel to [010]

    [ P222(m,n,o,p)], [P222_1(c,d)], [C222_1(b)], [C222(g,h)], [F222(\,f,i)], [I222(g,h)], [I2_12_12_1(b)], [Pmm2(e,f)], [Cmm2(d)], [Amm2(c)], [Abm2(c)], [Fmm2(d)], [Imm2(c)], [Pmmm(w,x)], [Pnnn(i,j)], [Pccm(k,l)], [Pban(i,j)], [Pmma(a,b,c,d,g,h,i,j)], [Pmna(g)], [Pcca(c)], [Pmmm(\,f)], [Pbcn], [Pnma], [Cmca(e)], [Cmmm(o)], [Cccm(h)], [Cmma (e,f,j,k,n)], [Ccca(\,f)], [Fmmm(d,k,n)], [Fddd(\,f)], [Immm(m)], [Ibam(g)], [Ibca(d)], [Imma(c,d,g,i)]

  • (c) β13 = β23 = 0; one principal axis parallel to [001]

    [P222(q,r,s,t)], [P2_12_12], [C222(i,j,k)], [F222(g,h)], [I222(i,j)], [I2_12_12_1(c)], [Pcc2], [Pma2(a,b)], [Pnc2], [Pba2], [Pnn2], [Cmm2(c)], [Ccc2], [ Abm2(a,b)], [Ama2(a)], [Aba2], [Fmm2(b)], [Fdd2], [Iba2], [Ima2(a)], [Pmmm(y,z)], [Pnnn(k,l)], [Pccm(a,b,c,d,m,n,o,p,q)], [Pban(k,l)], [ Pnna(c)], [Pcca(d,e)], [Pbam], [Pccn], [Pbcm(d)], [Pnnm], [Cmcm(g)], [Cmmm(e,f,m,p,q)], [Cccm(c,d,e,f,i,j,k,l)], [Cmma(l)], [Ccca(g,h)], [Fmmm(e,j,o)], [Fddd(g)], [Immm(n)], [Ibam(c,d,h,i,j)], [Ibca(e)]

(2) Site symmetry mm2, 222, mmm – three independent elements
  • (a) β12 = β13 = β23 = 0; principal axes parallel to crystal axes

    All space groups

(C) Tetragonal.

(1) Site symmetry m, 2, 2/m – four independent elements
  • (a) β12 = β13 = 0; one principal axis parallel to [100]

    [ P422(l,m,n,o)], [P4_222(\,j,k,l,m)], [I422(h,i)], [I4_122(\,f)], [I4_1md], [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)], [P4/mcc(k,l)], [P4/nbm(k,l)], [P4/nnc(i,j)], [P4/nmm(i)], [P4_2/mmc(o,p)], [P4_2/mcm(l,m)], [P4_2/nbc(h,i)], [P4_2/nnm(i,j)], [P4_2/nmc(g)], [ I4/mmm(n)], [I4/mcm(\,j)], [I4_1/amd(c,d,f,h)], [I4_1/acd(e)]

  • (b) β12 = β23 = 0; one principal axis parallel to [010]

    [P4_122(a,b)], [P4_322(a,b)], [P4mm(e,f)], [P4_2mc], [I4mm(d)], [P\overline {4}2c(h,j)], [P\overline {4}m2(\,j,k)], [I\overline {4}m2(i)], [P4/mmm(s,t)]

  • (c) β13 = β23 = 0; one principal axis parallel to [001]

    [P4], [P4_2], [I4], [I4_1], [P\overline {4}], [I\overline {4}], [P4/m], [P4_2/m], [P4/n], [P4_2/n], [I4/m], [I4_1/a], [P422(i)], [P42_12(d)], [P4_222(g,h,i)], [P4_22_12(c,d)], [I422(\,f)], [I4_122(c)], [P4_2cm(c)], [P4_2nm(b)], [P4cc], [P4nc], [P4_2bc], [I4_1cd], [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/mcc(e,i,m)], [P4/nnc(g)], [P4/mbm(i,j)], [P4/mnc(c,f,h)], [ P4/ncc(e)], [P42/mmc(q)], [P4_2/mcm(\,f,k,n)], [P4_2/nbc(\,f,g)], [P4_2/nnm(h)], [P4_2/mbc(a,c,e,f,h)], [P4_2/mnm(c,h,i)], [P4_2/ncm(\,f)], [I4/mmm(l)], [I4/mcm(k)], [I4_1/acd(d)]

  • (d) β11 = β22, β13 = −β23; one principal axis parallel to [110]

    [P422(\,j,k)], [P42_12(e,f)], [P4_122(c)], [P4_12_12], [P4_222(n,o)], [P4_22_12(e,f)], [P4_322(c)], [P4_32_12], [I422(g,j)], [I4_122(d)], [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/nbm(e,f,i,j,m)], [P4/nnc(h)], [P4/mnc(g)], [P4_2/mmc(n)], [P4_2/nbc(\,j)], [P4_2/nnm(e,f,k,l,m)], [P4_2/mbc(g)], [I4/mmm(k)], [I4/mcm(e,i)], [I4_1/amd(g)], [I4_1/acd(\,f)]

  • (e) β11 = β22, β13 = β23; one principal axis parallel to [[1\overline {1}0]]

    [I4_122(e)], [P4mm(d)], [P4bm], [P4_2cm(d)], [P4_2nm(c)], [I4mm(c)], [I4cm], [P\overline {4}2m(n)], [P\overline {4}2_1m(e)], [P\overline {4}n2(\,f)], [I\overline {4}2m(i)], [P4/mmm(r)], [P4/mbm(k)], [P4/nmm(d,e,g,h,j)], [P4/ncc(\,f)], [P4_2/mcm(o)], [P4_2/mnm(\,j)], [P4_2/nmc(\,f)], [P4_2/ncm(c,d,g,h,i)], [I4/mmm(\,f,m)], [I4/mcm(l)]

(2) Site symmetry mm2, 222, mmm – three independent elements
  • (a) β12 = β13 = β23 = 0; principal axes parallel to crystal axes

    [P422], [ P4_222(a,b,c,d)], [I422(c)], [P4mm], [P4_2mc], [ I4mm], [ I4_1md], [ P\overline {4}2m(e,f)], [ P\overline {4}2c], [ P\overline {4}m2], [ I\overline {4}m2], [ I\overline {4}2m(c)], [P4/mmm(e,f,i,l,m,n,o)], [P4/mcc], [P4/nnc], [P4/nmm], [P4_2/mmc], [P4_2/mcm(e)], [P4_2/nbc(a,b)], [P4_2/nnm(c)], [P4_2/nmc], [I4/mmm(c,g,i,j)], [I4_1/amd]

  • (b) β11 = β22, β13 = β23 = 0; principal axes parallel to [110], [[1\overline {1}0]] and [001]

    [P42_12], [P4_222(e,f)], [P4_22_12], [I422(d)], [I4_122], [P4bm], [P4_2cm], [P4_2nm], [I4cm], [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)], [P4/nbm], [P4/mbm], [P4/mnc], [P4/ncc], [P4_2/mcm(a,c,g,h,i,j)], [P4_2/nbc(c)], [P4_2/nnm(d,g)], [P4_2/mbc], [P4_2/mnm], [P4_2/ncm], [I4/mmm(h)], [I4/mcm], [I4_1/acd]

(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]

    [P6], [P6_2], [P6_4], [P\overline 6], [P6/m], [P6_3/m], [P622(i)], [P6_222(e,f)], [P6_422(e,f)], [P6cc], [P \overline 6m2(l,m)], [P\overline 6c2(k)], [P\overline 62m(\,j,k)], [P\overline 62c(h)], [P6/mmm(\,p,q)], [P6/mcc(g,i,l)], [P6_3/mcm(\,j)], [P6_3/mmc(\,j)]

  • (b) β11 = β22, β13 = −β23; one principal axis parallel to [110]

    [P3m1], [R3m], [P\overline 3m1(i)], [R\overline 3m(h)], [P6mm(e)], [P6_3mc], [P\overline 6m2(n)]

  • (c) β11 = β22, β13 = β23; one principal axis parallel to [[2\overline 10]]

    [P312], [P3_112], [P3_212], [P\overline 31m(i,j)], [P\overline 31c], [P622(l,m)], [P\overline 6c2(\;j)]

  • (d) β22 = 2β12, 2β13 = β23; one principal axis parallel to [100]

    [P321], [P3_121], [P3_221], [R32], [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)], [P6_122(a)], [P6_522(a)], [P6_222(g,h)], [P6_422(g,h)], [P6_322(g)], [P\overline {6}2c(g)], [P6/mmm(o)], [P6/mcc(\,j)], [P6_3/mmc(g,i,k)]

  • (e) β22 = 2β12, β23 = 0; one principal axis parallel to [210]

    [P31m], [P\overline {3}1m(\,f,g,k)], [P6_122(b)], [P6_522(b)], [P6_222(i,j)], [P6_422(i,j)], [P6_322(h)], [P6mm(d)], [P6_3cm], [P\overline {6}2m(i)], [P6/mmm(n)], [P6/mcc(k)], [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]

    [P622], [P6_222], [P6_422], [P6mm], [P\overline {6}2m], [P6/mmm], [P6/mcc], [P6_3/mcm], [P6_3/mmc]

  • (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]

    [P23], [F23], [I23], [I2_13], [Pm\overline {3}], [Pn\overline {3}], [Fm\overline {3}], [Fd\overline {3}], [Im\overline {3}], [Ia\overline {3}], [P432(h)], [P4_232(h,i,j)], [F432(i)], [F4_132(\,f)], [I432(g)], [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]

    [P432(i,j)], [P4_232(l)], [F432(g,h)], [I432(h)], [P4_132], [I4_132(g)], [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]]

    [ P4_232(k)], [F4_132(g)], [I432(i)], [P4_332], [I4_132(h)], [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

    [P23], [I23], [Pm\overline 3], [Pn\overline 3], [Fm\overline 3], [Im\overline 3], [P4_232(d)], [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]

    [P4_232(e,f)], [F432], [I432], [I4_132], [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