Local tensorial susceptibility of cubic crystals

Dmitrienko, V. E.; Kirfel, A.; Ovchinnikova, E. N.

doi:10.1107/97809553602060000910

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

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

Section 1.11.2.3. Local tensorial susceptibility of cubic crystals

V. E. Dmitrienko,^a ^* A. Kirfel^b and E. N. Ovchinnikova^c

^a A. 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: dmitrien@crys.ras.ru

1.11.2.3. 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) or atomic displacements (Chapter 1.9 of this volume) and electric field gradients (Chapter 2.2 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 $[R_{3}=\pmatrix{0&0&1\cr 1&0&0\cr 0&1&0}. \eqno(1.11.2.8)]$ This transformation results in the circular permutation $[x,y,z\to]$ [z,x,y] , and from equation (1.11.2.1) it is easy to see that invariance of $[\chi_{jk}(x,y,z)]$ demands the general form $[\chi_{jk}(x,y,z)=\pmatrix{ a_1(x,y,z)&a_2(z,x,y)&a_2(y,z,x)\cr a_2(z,x,y)&a_1(y,z,x)&a_2(x,y,z)\cr a_2(y,z,x)&a_2(x,y,z)&a_1(z,x,y)}, \eqno(1.11.2.9)]$ 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 $[a_i(x+\textstyle{1\over 2},y+\textstyle{1\over 2},z)]$ = $[a_i(x,y+\textstyle{1\over 2},z+\textstyle{1\over 2})]$ = $[a_i(x+\textstyle{1\over 2},y,z+\textstyle{1\over 2})]$ = 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] : $[\eqalignno{a_1(x,y,z)&=a_1(x,\bar{y},\bar{z})=a_1(\bar{x},\bar{y},z) =a_1(\bar{x},y,\bar{z}),& \cr a_2(x,y,z)&=a_2(x,\bar{y},\bar{z})=-a_2(\bar{x},\bar{y},z) =-a_2(\bar{x},y,\bar{z}).&\cr &&(1.11.2.10)}]$ [P2_13,I2_13] : $[\eqalignno{a_1(x,y,z)&=a_1(\textstyle{1\over 2}+x,\textstyle{1\over 2}-y,\bar{z}) &\cr &=a_1(\textstyle{1\over 2}-x,\bar{y},\textstyle{1\over 2}+z)=a_1(\bar{x},\textstyle{1\over 2}+y,\textstyle{1\over 2}-z), &\cr a_2(x,y,z)&=a_2(\textstyle{1\over 2}+x,\textstyle{1\over 2}-y,\bar{z}) &\cr&=-a_2(\textstyle{1\over 2}-x,\bar{y},\textstyle{1\over 2}+z)=-a_2(\bar{x},\textstyle{1\over 2}+y,\textstyle{1\over 2}-z). &\cr&&(1.11.2.11)}]$ $[Pm\bar{3}, Fm\bar{3}, Im\bar{3}]$ : (1.11.2.10) and $[a_i(x,y,z)=a_i(\bar{x},\bar{y},\bar{z}). \eqno(1.11.2.12)]$ $[Pn\bar{3}]$ : (1.11.2.10) and $[a_i(x,y,z)=a_i(\textstyle{1\over 2}-x,\textstyle{1\over 2}-y,\textstyle{1\over 2}-z). \eqno(1.11.2.13)]$ $[Fd\bar{3}]$ : (1.11.2.10) and $[a_i(x,y,z)=a_i(\textstyle{1\over 4}-x,\textstyle{1\over 4}-y,\textstyle{1\over 4}-z). \eqno(1.11.2.14)]$ $[Pa\bar{3}, Ia\bar{3}]$ : (1.11.2.11) and (1.11.2.12).

[P432, F432, I432] : (1.11.2.10) and $[a_i(x,y,z)=a_i(\bar{x},\bar{z},\bar{y}). \eqno(1.11.2.15)]$

[P4_232] : (1.11.2.10) and $[a_i(x,y,z)=a_i(\textstyle{1\over 2}-x,\textstyle{1\over 2}-z,\textstyle{1\over 2}-y).\eqno(1.11.2.16)]$ [F4_132, P4_332, I4_132] : (1.11.2.11) and $[a_i(x,y,z)=a_i(\textstyle{1\over 4}-x,\textstyle{1\over 4}-z,\textstyle{1\over 4}-y).\eqno(1.11.2.17)]$ [P4_132] : (1.11.2.11) and $[a_i(x,y,z)=a_i(\textstyle{3\over 4}-x,\textstyle{3\over 4}-z,\textstyle{3\over 4}-y).\eqno(1.11.2.18)]$ $[P\bar{4}3m, F\bar{4}3m, I\bar{4}3m]$ : (1.11.2.10) and $[a_i(x,y,z)=a_i(x,z,y).\eqno(1.11.2.19)]$ $[P\bar{4}3n, F\bar{4}3c]$ : (1.11.2.10) and $[a_i(x,y,z)=a_i(\textstyle{1\over 2}+x,\textstyle{1\over 2}+z,\textstyle{1\over 2}+y).\eqno(1.11.2.20)]$ $[I\bar{4}3d]$ : (1.11.2.11) and $[a_i(x,y,z)=a_i(\textstyle{1\over 4}+x,\textstyle{1\over 4}+z,\textstyle{1\over 4}+y).\eqno(1.11.2.21)]$ $[Pm\bar{3}m,Fm\bar{3}m,Im\bar{3}m]$ : (1.11.2.10), (1.11.2.12) and (1.11.2.19).

$[Pn\bar{3}n]$ : (1.11.2.10), (1.11.2.13) and (1.11.2.15).

$[Pm\bar{3}n, Fm\bar{3}c]$ : (1.11.2.10), (1.11.2.12) and (1.11.2.20).

$[Pn\bar{3}m]$ : (1.11.2.10), (1.11.2.13) and (1.11.2.19).

$[Fd\bar{3}m]$ : (1.11.2.10), (1.11.2.14) and (1.11.2.19).

$[Fd\bar{3}c]$ : (1.11.2.10), (1.11.2.13) and (1.11.2.20).

$[Ia\bar{3}d]$ : (1.11.2.11), (1.11.2.12) and (1.11.2.21).

For all [a_i(x,y,z)] , the sets of coordinates are chosen here as in International Tables for Crystallography Volume A (Hahn, 2005); the first one being adopted if Volume A offers two alternative origins. The expressions (1.11.2.10) or (1.11.2.11) 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 1.11.2.2. For the glide plane [c] , the tensor structure factor of $[0k\ell\semi\ell=2n+1]$ reflections is given by (1.11.2.6), whereas for the diagonal glide plane [n] , it is given by $[F_{jk}(hh\ell\semi\ell=2n+1)=\pmatrix{F_1&0&F_2\cr 0&-F_1&-F_2\cr F_2&-F_2&0},\eqno(1.11.2.22)]$ 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 $[F_{jk}(00\ell)=\pmatrix{0&0&F_1\cr 0&0&F_2\cr F_1&F_2&0}\eqno(1.11.2.23)]$ or $[F_{jk}(00\ell)=\pmatrix{F_1&F_2&0\cr F_2&-F_1&0\cr 0&0&0}, \eqno(1.11.2.24)]$ see Table 1.11.2.2.

Table 1.11.2.2 | top | pdf |
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 group	Indices of reflections	Expressions for $[F_{jk}(hk\ell)]$ and additional restrictions
	$[00\ell{:}\ \ell=2n+1]$	(1.11.2.23)
$[Pn\bar3]$	$[0k\ell{:}\ \ell=2n+1]$	(1.11.2.6); for $[00\ell]$
$[Fd\bar3]$	$[0k\ell{:}\ k,\ell=2n, k+\ell=4n+2]$	(1.11.2.6); for $[00\ell]$
$[Pa\bar3]$	$[0k\ell{:}\ k=2n+1]$	(1.11.2.6); for
$[Ia\bar3]$	$[0k\ell{:}\ k,\ell=2n+1]$	(1.11.2.6)
	$[00\ell{:}\ \ell=2n+1]$	(1.11.2.24)
	$[00\ell{:}\ \ell=4n+2]$	(1.11.2.24)
	$[00\ell{:}\ \ell=4n\pm 1]$	(1.11.2.23); $[F_2=\mp iF_1]$
	$[00\ell{:}\ \ell=4n+2]$	(1.11.2.24)
	$[00\ell{:}\ \ell=4n\pm 1]$	(1.11.2.23); $[F_2=\pm iF_1]$
	$[00\ell{:}\ \ell=4n+2]$	(1.11.2.24)
	$[00\ell{:}\ \ell=4n+2]$	(1.11.2.24)
$[P\bar43n]$	$[hh\ell{:}\ \ell=2n+1]$	(1.11.2.22); for $[00\ell]$ , for
$[F\bar43c]$	$[hh\ell{:}\ h,\ell=2n+1]$	(1.11.2.22); for
$[I\bar43d]$	$[hh\ell{:}\ 2h+\ell=4n+2]$	(1.11.2.22); for $[00\ell]$ , for
$[Pn\bar3n]$	$[hh\ell{:}\ \ell=2n+1]$	(1.11.2.22); for
	$[0k\ell{:}\ k+\ell=2n+1]$	(1.11.2.6); for $[00\ell]$
$[Pm\bar3n]$	$[hh\ell{:}\ \ell=2n+1]$	(1.11.2.22); for
$[Pn\bar3m]$	$[0k\ell{:}\ k+\ell=2n+1]$	(1.11.2.6); for $[00\ell]$
$[Fm\bar3c]$	$[hh\ell{:}\ h,\ell=2n+1]$	(1.11.2.22); for
$[Fd\bar3m]$	$[0k\ell{:}\ k,\ell=2n,k+\ell=4n+2]$	(1.11.2.6); for $[00\ell]$
$[Fd\bar3c]$	$[0k\ell{:}\ k,\ell=2n,k+\ell=4n+2]$	(1.11.2.6); for $[00\ell]$
	$[hh\ell{:}\ h,\ell=2n+1]$	(1.11.2.22); for
$[Ia\bar3d]$	$[0k\ell{:}\ k,\ell=2n+1]$	(1.11.2.6); for
	$[hh\ell{:}\ 4h+\ell=4n+2]$	(1.11.2.22); : , for $[00\ell]$