Elasticity tensor for an icosahedral 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. 255

Section 1.10.4.6.5. Elasticity tensor for an icosahedral quasicrystal

T. Janssen^a ^*

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

1.10.4.6.5. Elasticity tensor for an icosahedral quasicrystal

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The point group of an icosahedral quasicrystal is 532(5²32) with generators having components $[\displaylines{A_E =\pmatrix{1&\tau&-1-\tau\cr\tau&1+\tau&1\cr1+\tau&-1&\tau\cr} /2, \cr B_E = \pmatrix{-\tau&1+\tau&-1\cr1+\tau&1&\tau\cr 1&-\tau&-1-\tau\cr}/2 }]$ in physical space and components $[A_I =\pmatrix{-1&\tau&-1-\tau\cr -\tau&\tau^{-1}&1\cr1+\tau&1&-\tau\cr} /2,\quad B_I = \pmatrix{0&0&-1\cr-1&0&0 \cr0&1&0\cr}]$ in internal space (see Table 1.10.5.2). The phonon and phason strain tensors form a 6D, respectively 9D, vector space, in which the point group acts with matrices $[\displaylines{S_e ={1\over4}\pmatrix{1&\tau\sqrt{2}&-\varphi\sqrt{2}&1-\tau&-\sqrt{2}&2+\tau\cr \tau\sqrt{2}&2&0&\sqrt{2}&-2&-\varphi\sqrt{2}\cr \varphi\sqrt{2}&0&-2&-\tau\sqrt{2}&2&-\sqrt{2}\cr 1-\tau&\sqrt{2}&\tau\sqrt{2}&2+\tau&\varphi\sqrt{2}&1\cr \sqrt{2}&2&2&-\varphi\sqrt{2}&0&\tau\sqrt{2}\cr 2+\tau &-\varphi\sqrt{2} &\sqrt{2}&1& -\tau\sqrt{2}& 1-\tau\cr},\cr S_f ={1\over4}\pmatrix{-\tau&\varphi&-1&\tau -1&1&-\tau &1&-2-\tau&\varphi\cr \varphi &1&\tau&1&\tau&1-\tau&-2-\tau&-\varphi&-1\cr\ 1&-\tau&-\varphi&\tau &\tau -1&-1&-\varphi&1&2+\tau\cr \tau -1&1&-\tau &-1&2+\tau&-\varphi&-\tau&\varphi&-1\cr 1&\tau&1-\tau& 2+\tau&\varphi&1&\varphi&1&\tau\cr \tau&\tau -1&-1&\varphi &-1&-2-\tau & 1&-\tau&-\varphi\cr -1&2+\tau&-\varphi&\tau&-\varphi&1&\tau -1&1&-\tau\cr 2+\tau &\varphi &1 & -\varphi& -1& -\tau&1 &\tau&1-\tau\cr \varphi& -1&-2-\tau&-1&\tau&\varphi& \tau & \tau -1& -1\cr}}]$ and $[\displaylines{T_e ={1\over4}\pmatrix{1&-\tau\sqrt{2}&\varphi\sqrt{2}&1-\tau&-\sqrt{2}&2+\tau\cr \tau\sqrt{2}&-2&0&\sqrt{2} &-2&-\varphi\sqrt{2}\cr -\varphi\sqrt{2} & 0& -2& \tau\sqrt{2}& -2& \sqrt{2}\cr 1-\tau& -\sqrt{2}& -\tau\sqrt{2} & 2+\tau& \varphi\sqrt{2}&1\cr -\sqrt{2}& 2&2&\varphi\sqrt{2}&0&-\tau\sqrt{2}\cr 2+\tau&\varphi\sqrt{2}&-\sqrt{2}&1& -\tau\sqrt{2}&1-\tau\cr},\cr T_f ={1\over2}\pmatrix{0&0&1&0&0&-\tau&0&0&\varphi\cr 1&0&0&-\tau&0&0&\varphi&0&0\cr 0&-1&0&0&\tau&0&0&-\varphi&0\cr 0&0&\tau&0&0&-\varphi&0&0&-1\cr \tau&0&0&-\varphi&0&0&-1&0&0\cr 0&-\tau&0&0&\varphi&0&0&1&0\cr 0&0&-\varphi&0&0&-1&0&0&\tau\cr -\varphi&0&0&-1&0&0&\tau&0&0\cr 0&\varphi&0&0&1&0&0&-\tau&0\cr}.}]$ This implies that the phonon elasticity tensors form a 21D space, the phason elasticity tensors a 45D space and the phonon–phason coupling a 54D space. The invariant vectors under these orthogonal transformations correspond to invariant elastic tensors. Their coordinates are the elastic constants. For the given presentation of the point group, these are given in Table 1.10.4.3. The tensor elements are expressed in parameters x and y where there are two independent tensor elements. The tensor elements that are not given are zero or equal to that given by the permutation symmetry. If bases for the phonon and phason strain are introduced by [[1]=11, [2]=12, [3]=13, [4]=22, [5]=23, [6]=33] for the phonon part and $[\displaylines{[1]=14, [2]=15, [3]=16, [4]=24, [5]=25, [6]=26, \cr [7]=34, [8]=35, [9]=36}]$ for the phason part, the elastic tensors may be given in matrix form as $[\displaylines{c^{ee}=\pmatrix{x+y&0&0&x&0&x\cr 0&y&0&0&0&0\cr 0&0&y&0&0&0\cr x&0&0&x+y&0&x\cr 0&0&0&0&y&0\cr x&0&0&x&0&x+y\cr}, \cr c^{ff}=\pmatrix{z&\tau^2 u&- \tau u& - \tau u& \tau u&- \tau u&-u&0& \tau u \cr \tau^2 u&z- 2\tau u&u& \tau u&u&0&0&\tau^2 u& \tau u \cr - \tau u &u& z&- \tau u&0&-\tau^2 u& \tau u& \tau u& \tau u\cr - \tau u& \tau u&-\tau u&z& \tau u&u&-\tau^2 u& \tau u&0\cr \tau u&u&0& \tau u&z&-\tau^2 u& \tau u&- \tau u&- \tau u\cr - \tau u&0&-\tau^2 u&u&-\tau^2 u&z-2 \tau u&0&- \tau u&u\cr -u&0& \tau u&-\tau^2 u& \tau u&0&z-2 \tau u&-u&-\tau^2 u\cr 0&\tau^2 u& \tau u& \tau u&- \tau u&- \tau u&-u&z&- \tau u\cr \tau u& \tau u& \tau u&0&- \tau u&u&-\tau^2 u&- \tau u &z\cr}, \cr c^{ef}=\pmatrix{-v&- \tau v&-\tau^2 v&-\tau^3v& \tau v&-\tau^2 v&v&- \tau v&-\tau^{-1}v\cr -\tau^3v& \tau v&-\tau^2 v&\tau^2v&v& \tau v&0&0&0\cr v&- \tau v&-\tau^{-1}v&0&0&0&- \tau v&-\tau^2 v&-\tau^3 v\cr \tau^{-1}v&v& \tau v&-\tau^2 v&v&- \tau v&-\tau^2v&-\tau^3 v& \tau v\cr\ 0&0&0&-\tau^2 v&\tau^3 v& \tau v& \tau v&-\tau^{-1} v&v\cr -\tau v&-\tau^2v&-\tau^3v& \tau v&-\tau^{-1} v&v&- \tau v&\tau^2 v&v \cr}.}]$

Table 1.10.4.3| top | pdf |
Elastic constants for icosahedral quasicrystals

Type	Free parameters	Relations
Phonon–phonon	2	$[ c_{1111}=c_{2222}=c_{3333}=x]$
		$[c_{1122}=c_{1133}=c_{2233}=y]$
		$[c_{1212}=c_{1313}=c_{2323}=x-y]$
Phason–phason	2	$[c_{1414}=c_{1616}=c_{2424}=c_{2525}=c_{3535}=c_{3636}=x]$
		$[c_{1416}=c_{1424}=-c_{1425}=c_{1426}=-c_{1436}=y]$
		$[c_{1524}=c_{1536}=-c_{1624}=c_{1634}=c_{1635}=-y]$
		$[c_{1636}=c_{2425}=c_{2435}=c_{2534}=-c_{2535}=-y]$
		$[c_{2536}=c_{2635}=c_{3536}=y, c_{1515}=c_{2626}=c_{3434}=x+2y]$
		$[c_{1415}=c_{1535}=-c_{1626}=-c_{2434}=-c_{2526}=-c_{3436}=y\tau]$
		$[c_{1434}=-c_{11516}=-c_{1525}=-c_{2426}=-c_{2636}=c_{3435}=y/\tau]$
Phonon–phason	1	$[-c_{1114}=c_{1134}=c_{1225}=c_{2225}=c_{2336}=c_{3326}=c_{3336}]$
		$[\quad=-c_{1115}/\tau =c_{1125}/\tau =-c_{1135}\tau=c_{1215}\tau =c_{1226}\tau]$
		$[\quad=-c_{1334}\tau =-c_{2226}\tau =c_{2236}\tau =c_{2326}\tau =c_{2334}\tau ]$
		$[\quad=-c_{3334}\tau =-c_{1116}\tau^2 =-c_{1126}\tau^2 =-c_{1216}\tau^2 ]$
		$[\quad=-c_{1335}\tau^2 =-c_{2224}\tau^2 =-c_{2234}\tau^2 =c_{3335}\tau^2 ]$
		$[\quad=-c_{1136}\tau =c_{1224}\tau =-c_{1316}\tau =-c_{2335}\tau ]$
		$[\quad=-c_{1124}/\tau^3 =-c_{1336}/\tau^3 =c_{2235}/\tau^3 =c_{2325}/\tau^3 ]$