Elasticity tensor for a two-dimensional octagonal quasicrystal

Janssen, T.

doi:10.1107/97809553602060000637

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

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International Tables for Crystallography (2006). Vol. D. ch. 1.10, p. 254

Section 1.10.4.6.3. Elasticity tensor for a two-dimensional octagonal quasicrystal

T. Janssen^a ^*

^a Institute for Theoretical Physics, University of Nijmegen, 6524 ED Nijmegen, The Netherlands
Correspondence e-mail: ted@sci.kun.nl

1.10.4.6.3. Elasticity tensor for a two-dimensional octagonal quasicrystal

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The point group of the standard octagonal tiling is generated by the 2D orthogonal matrices $[A = \pmatrix{\sqrt{1/2}&-\sqrt{1/2}\cr \sqrt{1/2}&\sqrt{1/2}\cr},\quad B = \pmatrix{-1&0\cr 0&1\cr}. ]$ In the tensor space one has the following transformations of the basis vectors; they are denoted by ij for $[{\bf e}_i\otimes{\bf e}_j]$ : $[\eqalign{11 &\rightarrow {\textstyle{1\over2}}(11+12+21+22)\cr 12 &\rightarrow {\textstyle{1\over2}}(-11+12-21+22)\cr 21 &\rightarrow {\textstyle{1\over2}}(-11 - 12+21+22)\cr 22 &\rightarrow {\textstyle{1\over2}}(11 -12-21+22).\cr}]$ In the space spanned by a = 11, $[b=\sqrt{1/2}(12+21)]$ and c = 22, the eightfold rotation is represented by the matrix $[S_E = \pmatrix{{1\over2}&-\sqrt{1/2}&{1\over2} \cr \sqrt{1/2}&0&-\sqrt{1/2} \cr {1\over2}&\sqrt{1/2}&{1\over2} \cr}.]$ In the six-dimensional space with basis aa, $[\sqrt{1/2}(ab+ba)]$ , $[\sqrt{1/2}(ac+ca)]$ , bb, $[\sqrt{1/2}(bc+cb)]$ and cc, the rotation gives the transformation $[\pmatrix{{1\over4}&{1\over2}&\sqrt{2}/4&{1\over2}&{1\over2}&{1\over4}\cr -{1\over2} &-{1\over2}&0&0&{1\over2}&{1\over2}\cr \sqrt{2}/4&0&{1\over2}&-\sqrt{1/2}&0&\sqrt{2}/4\cr {1\over2}&0&-\sqrt{1/2}&0&0&{1\over2}\cr -{1\over2}&{1\over2}&0&0&-{1\over2}&{1\over2}\cr {1\over4}&-{1\over2}&\sqrt{2}/4&{1\over2}&-{1\over2}&{1\over4}\cr}.]$ The vector $[{\bf v}]$ such that $[{\bf S}\cdot {\bf v}={\bf v}]$ then is of the form $[ {\bf v} = (v_1, 0, (v_1 -v_4)\sqrt{2}, v_4, 0, v_1)^{T}. ]$ This vector is also invariant under the mirror B. This means that there are two independent phonon elastic constants $[c_{1111}^{E}]$ and $[c_{1212}^{E}]$ , whereas the other tensor elements satisfy the relations $[\displaylines{c_{1112}^{E} = c_{1222}^{E} = 0,\quad c_{2222}^{E} = c_{1111}^{E}, \cr c_{1122}^{E} = (c_{1111}^{E} + c_{1212}^{E})\sqrt{1/2}.}]$ The internal component of the eightfold rotation is A³, that of the mirror B is B itself. The phason strain tensor transforms with the tensor product of external and internal components. This implies that the basis vectors, denoted by ij (i = 1, 2; j = 3, 4), transform under the eightfold rotation according to $[\eqalign{13 &\rightarrow (-13+14 - 23+24)/2\cr 14 &\rightarrow (-13 - 14-23-24)/2\cr 23 &\rightarrow (13-14-23+24)/2\cr 24 &\rightarrow (13+14-23-24)/2.\cr}]$ The symmetrized tensor square of this matrix gives the transformation in the space of phason–phason elasticity tensors, the direct product of the transformations in the 3D phonon strain space and the 4D phason strain space gives the transformation in the space of phonon–phason elasticity tensors. The first matrix is given by $[{1\over16}\pmatrix{1&\sqrt{2}&-\sqrt{2}&-\sqrt{2}&1&-\sqrt{2}&-\sqrt{2}&1&\sqrt{2}&1\cr -\sqrt{2}&0&2&0&\sqrt{2}&0&-2&-\sqrt{2}&0&\sqrt{2}\cr \sqrt{2}&2&0&0&\sqrt{2}&0&0&-\sqrt{2}&-2&-\sqrt{2}\cr -\sqrt{2}&0&0&2&\sqrt{2}&-2&0&\sqrt{2}&0&-\sqrt{2}\cr 1&-\sqrt{2}&-\sqrt{2}&\sqrt{2}&1&\sqrt{2}&-\sqrt{2}&1&-\sqrt{2}&1\cr -\sqrt{2}&0&0&-2&\sqrt{2}&2&0&\sqrt{2}&0&-\sqrt{2}\cr \sqrt{2}&-2&0&0&\sqrt{2}&0&0&-\sqrt{2}&2&-\sqrt{2}\cr 1&\sqrt{2}&\sqrt{2}&\sqrt{2}&1&\sqrt{2}&\sqrt{2}&1&\sqrt{2}&1\cr-\sqrt{2}&0&-2&0&\sqrt{2}&0&2&-\sqrt{2}&0&\sqrt{2}\cr 1&-\sqrt{2}&\sqrt{2}&-\sqrt{2}&1&-\sqrt{2}&\sqrt{2}&1&-\sqrt{2}&1\cr}. ]$ Vectors invariant under this operation and the transformation corresponding to the mirror B correspond to invariant elasticity tensors. For the transformation B, all tensor elements with an odd number of indices 1 or 3 are zero. In the space of phason–phason tensors the general invariant vector is $[ (x_1,0,0,-x_6+(x_5 -x_1)\sqrt{2},x_5,x_6,0,x_5,0,x_1). ]$

There are three independent elastic constants, $[x_1=c_{1313}]$ , $[x_5=c_{1414}]$ and $[x_6=c_{1423}]$ . For the phonon–phason elastic constants the corresponding invariant vector is $[(x,0,0,x,0,x/\sqrt{2},-x/\sqrt{2},0,-x,0,0,-x). ]$ The independent elastic constant is x = $[c_{1113}]$ = $[c_{1124}]$ = $[c_{1214}\sqrt{2}]$ = $[-c_{1223}\sqrt{2}]$ = $[-c_{2213}]$ = $[-c_{2224}]$ .

References

International Tables for Crystallography (2006). Vol. D. ch. 1.10, p. 254