Tables for
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.11, pp. 271-272

Section Local tensorial susceptibility of cubic crystals

V. E. Dmitrienko,a* A. Kirfelb and E. N. Ovchinnikovac

aA. V. Shubnikov Institute of Crystallography, Leninsky pr. 59, Moscow 119333, Russia,bSteinmann Institut der Universität Bonn, Poppelsdorfer Schloss, Bonn, D-53115, Germany, and cFaculty of Physics, M. V. Lomonosov Moscow State University, Leninskie Gory, Moscow 119991, Russia
Correspondence e-mail: Local tensorial susceptibility of cubic crystals

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Let us consider in more detail the local tensorial properties of cubic crystals. This case is particularly interesting because for cubic symmetry the second-rank tensor is isotropic, so that a global anisotropy is absent (but it exists for tensors of rank 4 and higher). Local anisotropy is of importance for some physical parameters, and it can be described by tensors depending periodically on the three space coordinates. This does not only concern X-ray susceptibility, but can also, for instance, result from describing orientation distributions in chiral liquid crystals (Belyakov & Dmitrienko, 1985[link]) or atomic displacements (Chapter 1.9[link] of this volume) and electric field gradients (Chapter 2.2[link] of this volume) in conventional crystals.

The symmetry element common to all cubic space groups is the threefold axis along the cube diagonal. The matrix [R_{3}] of the symmetry operation is[R_{3}=\pmatrix{0&0&1\cr 1&0&0\cr 0&1&0}. \eqno(]This transformation results in the circular permutation [x,y,z\to] [z,x,y], and from equation ([link] it is easy to see that invariance of [\chi_{jk}(x,y,z)] demands the general form[\chi_{jk}(x,y,z)=\pmatrix{ a_1(x,y,z)&a_2(z,x,y)&a_2(y,z,x)\cr a_2(z,x,y)&a_1(y,z,x)&a_2(x,y,z)\cr a_2(y,z,x)&a_2(x,y,z)&a_1(z,x,y)}, \eqno(]where [a_1(x,y,z)] and [a_2(x,y,z)] are arbitrary functions with the periodicity of the corresponding Bravais lattice: [a_i(x+n_x,y+n_y,z+n_z)=a_i(x,y,z)] for primitive lattices ([n_x,n_y,n_z] being arbitrary integers) plus in addition [a_i(x+\textstyle{1\over 2},y+\textstyle{1\over 2},z+\textstyle{1\over 2})] = [a_i(x,y,z)] for body-centered lattices or [a_i(x+\textstyle{1\over 2},y+\textstyle{1\over 2},z)] = [a_i(x,y+\textstyle{1\over 2},z+\textstyle{1\over 2})] = [a_i(x+\textstyle{1\over 2},y,z+\textstyle{1\over 2})] = [a_i(x,y,z)] for face-centered lattices.

Depending on the space group, other symmetry elements can enforce further restrictions on [a_1(x,y,z)] and [a_2(x,y,z)]:

[P23,F23,I23]:[\eqalignno{a_1(x,y,z)&=a_1(x,\bar{y},\bar{z})=a_1(\bar{x},\bar{y},z) =a_1(\bar{x},y,\bar{z}),& \cr a_2(x,y,z)&=a_2(x,\bar{y},\bar{z})=-a_2(\bar{x},\bar{y},z) =-a_2(\bar{x},y,\bar{z}).&\cr &&(}][P2_13,I2_13]:[\eqalignno{a_1(x,y,z)&=a_1(\textstyle{1\over 2}+x,\textstyle{1\over 2}-y,\bar{z}) &\cr &=a_1(\textstyle{1\over 2}-x,\bar{y},\textstyle{1\over 2}+z)=a_1(\bar{x},\textstyle{1\over 2}+y,\textstyle{1\over 2}-z), &\cr a_2(x,y,z)&=a_2(\textstyle{1\over 2}+x,\textstyle{1\over 2}-y,\bar{z}) &\cr&=-a_2(\textstyle{1\over 2}-x,\bar{y},\textstyle{1\over 2}+z)=-a_2(\bar{x},\textstyle{1\over 2}+y,\textstyle{1\over 2}-z). &\cr&&(}][Pm\bar{3}, Fm\bar{3}, Im\bar{3}]: ([link] and[a_i(x,y,z)=a_i(\bar{x},\bar{y},\bar{z}). \eqno(][Pn\bar{3}]: ([link] and[a_i(x,y,z)=a_i(\textstyle{1\over 2}-x,\textstyle{1\over 2}-y,\textstyle{1\over 2}-z). \eqno(][Fd\bar{3}]: ([link] and[a_i(x,y,z)=a_i(\textstyle{1\over 4}-x,\textstyle{1\over 4}-y,\textstyle{1\over 4}-z). \eqno(][Pa\bar{3}, Ia\bar{3}]: ([link] and ([link].

[P432, F432, I432]: ([link] and[a_i(x,y,z)=a_i(\bar{x},\bar{z},\bar{y}). \eqno(]

[P4_232]: ([link] and[a_i(x,y,z)=a_i(\textstyle{1\over 2}-x,\textstyle{1\over 2}-z,\textstyle{1\over 2}-y).\eqno(][F4_132, P4_332, I4_132]: ([link] and[a_i(x,y,z)=a_i(\textstyle{1\over 4}-x,\textstyle{1\over 4}-z,\textstyle{1\over 4}-y).\eqno(][P4_132]: ([link] and[a_i(x,y,z)=a_i(\textstyle{3\over 4}-x,\textstyle{3\over 4}-z,\textstyle{3\over 4}-y).\eqno(][P\bar{4}3m, F\bar{4}3m, I\bar{4}3m]: ([link] and[a_i(x,y,z)=a_i(x,z,y).\eqno(][P\bar{4}3n, F\bar{4}3c]: ([link] and[a_i(x,y,z)=a_i(\textstyle{1\over 2}+x,\textstyle{1\over 2}+z,\textstyle{1\over 2}+y).\eqno(][I\bar{4}3d]: ([link] and[a_i(x,y,z)=a_i(\textstyle{1\over 4}+x,\textstyle{1\over 4}+z,\textstyle{1\over 4}+y).\eqno(][Pm\bar{3}m,Fm\bar{3}m,Im\bar{3}m]: ([link], ([link] and ([link].

[Pn\bar{3}n]: ([link], ([link] and ([link].

[Pm\bar{3}n, Fm\bar{3}c]: ([link], ([link] and ([link].

[Pn\bar{3}m]: ([link], ([link] and ([link].

[Fd\bar{3}m]: ([link], ([link] and ([link].

[Fd\bar{3}c]: ([link], ([link] and ([link].

[Ia\bar{3}d]: ([link], ([link] and ([link].

For all [a_i(x,y,z)], the sets of coordinates are chosen here as in International Tables for Crystallography Volume A (Hahn, 2005[link]); the first one being adopted if Volume A offers two alternative origins. The expressions ([link] or ([link] appear for all space groups because all of them are supergroups of [P23] or [P2_13].

The tensor structure factors of forbidden reflections can be further restricted by the cubic symmetry, see Table[link]. For the glide plane [c], the tensor structure factor of [0k\ell\semi\ell=2n+1] reflections is given by ([link], whereas for the diagonal glide plane [n], it is given by[F_{jk}(hh\ell\semi\ell=2n+1)=\pmatrix{F_1&0&F_2\cr 0&-F_1&-F_2\cr F_2&-F_2&0},\eqno(]and additional restrictions on [F_1] and [F_2] can become effective for [k=\ell] or [h=\ell]. For forbidden reflections of the [00\ell] type, the tensor structure factor is either[F_{jk}(00\ell)=\pmatrix{0&0&F_1\cr 0&0&F_2\cr F_1&F_2&0}\eqno(]or[F_{jk}(00\ell)=\pmatrix{F_1&F_2&0\cr F_2&-F_1&0\cr 0&0&0}, \eqno(]see Table[link].

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The indices of the forbidden reflections and corresponding tensors of structure factors [F_{jk}(hk\ell)] for the cubic space groups ([n = 0, \pm 1, \pm 2,\ldots])

Space groupIndices of reflectionsExpressions for [F_{jk}(hk\ell)] and additional restrictions
[P2_13] [00\ell{:}\ \ell=2n+1] ([link]
[Pn\bar3] [0k\ell{:}\ \ell=2n+1] ([link]; [F_2=0] for [00\ell]
[Fd\bar3] [0k\ell{:}\ k,\ell=2n, k+\ell=4n+2] ([link]; [F_2=0] for [00\ell]
[Pa\bar3] [0k\ell{:}\ k=2n+1] ([link]; [F_2=0] for [0k0]
[Ia\bar3] [0k\ell{:}\ k,\ell=2n+1] ([link]
[P4_232] [00\ell{:}\ \ell=2n+1] ([link]
[F4_132] [00\ell{:}\ \ell=4n+2] ([link]
[P4_332] [00\ell{:}\ \ell=4n\pm 1] ([link]; [F_2=\mp iF_1]
  [00\ell{:}\ \ell=4n+2] ([link]
[P1_332] [00\ell{:}\ \ell=4n\pm 1] ([link]; [F_2=\pm iF_1]
  [00\ell{:}\ \ell=4n+2] ([link]
[I4_132] [00\ell{:}\ \ell=4n+2] ([link]
[P\bar43n] [hh\ell{:}\ \ell=2n+1] ([link]; [F_2=0] for [00\ell], [F_1=F_2=0] for [hhh]
[F\bar43c] [hh\ell{:}\ h,\ell=2n+1] ([link]; [F_1=F_2=0] for [hhh]
[I\bar43d] [hh\ell{:}\ 2h+\ell=4n+2] ([link]; [F_2=0] for [00\ell], [F_1=F_2=0] for [hhh]
[Pn\bar3n] [hh\ell{:}\ \ell=2n+1] ([link]; [F_1=F_2=0] for [hhh]
  [0k\ell{:}\ k+\ell=2n+1] ([link]; [F_1=F_2=0] for [00\ell]
[Pm\bar3n] [hh\ell{:}\ \ell=2n+1] ([link]; [F_1=F_2=0] for [hhh]
[Pn\bar3m] [0k\ell{:}\ k+\ell=2n+1] ([link]; [F_2=0] for [00\ell]
[Fm\bar3c] [hh\ell{:}\ h,\ell=2n+1] ([link]; [F_1=F_2=0] for [hhh]
[Fd\bar3m] [0k\ell{:}\ k,\ell=2n,k+\ell=4n+2] ([link]; [F_2=0] for [00\ell]
[Fd\bar3c] [0k\ell{:}\ k,\ell=2n,k+\ell=4n+2] ([link]; [F_2=0] for [00\ell]
  [hh\ell{:}\ h,\ell=2n+1] ([link]; [F_1=F_2=0] for [hhh]
[Ia\bar3d] [0k\ell{:}\ k,\ell=2n+1] ([link]; [F_2=-F_1] for [0kk]
  [hh\ell{:}\ 4h+\ell=4n+2] ([link]; [hhh]: [F_1=F_2=0], [F_2=0] for [00\ell]


Belyakov, V. A. & Dmitrienko, V. E. (1985). The blue phase of liquid crystals. Sov. Phys. Usp. 28, 535–562.
Hahn, Th. (2005). Editor. International Tables for Crystallography, Volume A, Space-Group Symmetry, 5th ed. Heidelberg: Springer.

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