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Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals
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Rep. Prog. Phys.63 (2000) 1–39. Printed in the UK PII: S0034-4885(00)83737-9
Symmetry groups, physical property tensors, elasticity anddislocations in quasicrystals
Chengzheng Hu, Renhui Wang and Di-Hua DingDepartment of Physics, Wuhan University, Wuhan 430072, China
Received 19 January 1999, in final form 29 June 1999
Abstract
The first quasicrystal (QC) structure was observed in 1984. QCs possess long-rangeorientational and translational order while lacking the periodicity of crystals. An overview isgiven on some physical properties of QCs. It begins with group theory and symmetry. Thenthe thermodynamics of equilibrium properties and physical property tensors are discussed.Finally, the generalized elasticity theory of QCs and the elasticity theory of dislocations inQCs are presented.
0034-4885/00/010001+39$90.00 © 2000 IOP Publishing Ltd 1
2 C Z Hu et al
Contents
Page1. Introduction 32. Group theory and symmetry 4
2.1. Higher-dimensional description and symmetry operations 42.2. Symmetry groups 62.3. Experimental determination of symmetry groups of QCs 82.4. Symmetry breaking and structural transitions 9
3. Physical property tensors 113.1. Thermodynamics of equilibrium properties 113.2. Determination of non-vanishing independent components 133.3. Elastic constant tensors 153.4. Thermal expansion and piezoelectricity tensors 20
4. Generalized elasticity theory 214.1. Phonon and phason fields 214.2. General expressions for elastic equations 23
5. Elasticity theory of dislocation 265.1. Dislocations in QCs 265.2. Elastic fields induced by general dislocations 285.3. Elastic fields induced by straight dislocations 30Acknowledgments 34Appendix. Character analysis 34References 36
Physical properties of quasicrystals 3
1. Introduction
The discovery by Schechtmanet al (1984) that certain Al–Mn alloys exhibit a phase withnoncrystallographic icosahedral symmetry has aroused a great deal of excitement becauseit challenges long-held beliefs about the nature of translational order. Levine and Steinhart(1984) introduced the term quasicrystals (QCs) for these unusual systems exhibiting diffractionpatterns with noncrystallographic symmetry. Since this first experiment new kinds of QChave been subsequently observed. They display the decagonal symmetry found in Al-basedalloys (Bendersky 1985, Chattopadhyayet al 1985, Funget al 1986, Urbanet al 1986), thedodecagonal symmetry found in NiCr (Ishimasaet al1988), the octagonal symmetry found inNiSi (Wanget al 1987) and the pentagonal symmetry found in the alloy AlCoNiTb (Liet al1996). These new QCs are called two-dimensional (2D) QCs. They have a periodicity alongone axis (8-, 10-, 12-, or 5-fold axis) but quasiperiodicity in the 2D plane perpendicular to it.Besides these QCs there are also one-dimensional (1D) QCs, which consist of a quasiperiodicstacking of layers, each exhibiting a 2D crystal structure. There are a number of reports onsuch structures available in the literature (Toddet al 1986, Heet al 1988, Terauchiet al 1988,Tsaiet al 1992, Yanget al 1996).
Parallel to the significant progress in experimental work, theoretical studies have alsobeen developed. There are two main approaches to such systems. The first approach utilizesmicroscopic models. The root of the construction of these microscopic models of QC structuresis Penrose tiling (Penrose 1974, 1979). A mathematical theory for Penrose tiling was providedby de Bruijn (1981) and the possibility of extending Penrose tiling to icosahedral symmetry hasbeen considered by several groups (Levine and Steinhardt 1984, 1986, Kramer and Neri 1984,Socolar and Steinhardt 1986, Socolaret al1986). A theoretical explanation of the existence ofBragg peaks in such a system was given by Levine and Steinhardt (1984, 1986). A very popularmethed for generating QCs is the projection formalism (Duneau and Katz 1985, Grunbauinand Shephard 1987). According to this theory, a QC structure in the physical space (VE) canbe obtained by projecting, ontoVE , a special set of points belonging to ann-dimensional (nD)lattice inside a strip (n > 3). A non-crystallographic symmetry group given in theVE can beembedded into a crystallographic group innD space and this embedding proves to be crucialfor the construction of quasilattices.nD crystallography was introduced by de Wolff (1974) todescribe a modulated structure in four-dimensional (4D) space and extended by Bak (1985a)and Janssen (1986) to include QCs. The subject of higher-dimensional symmetry groups hasbeen already expounded upon in the literature (Bieberbach 1910, Koster 1957, Brownet al1978, Janner and Janssen 1977, 1979, 1980, Schwarzenberger 1980, de Wolffet al1981). Thesecond approach utilizes continuum models based on the Landau theory. Such models providea powerful tool for studying the stability, phase transitions and physical properties of orderedphases. In the 1970s the Landau theory approach was employed to investigate the possiblecrystallographic structures of the solid below the phase transition (Baymet al1971, Alexanderand McTague 1978). Recently, it has been extended to quasicrystalline structures (Bak 1985b,Troian and Mermin 1985, Kaluginet al 1985, Jaric 1985, Bihamet al 1986, Widomet al1987).
This review is intended to provide an overview of some of the physical properties of QCs.The following section is devoted to group theory and symmetry. In section 3 we discuss thethermodynamics of equilibrium properties and physical property tensors. Sections 4 and 5present the generalized elasticity theory of QCs and the elasticity theory of dislocations inQCs.
4 C Z Hu et al
2. Group theory and symmetry
2.1. Higher-dimensional description and symmetry operations
The symmetry description for a quasiperiodic structure (QS) may be given by an embeddinginto a higher-dimensional space (Bak 1985a, Janssen 1986). In other words, a QS in ad-dimensional (dD) subspace (the physical space)VE can be obtained by intersecting a lattice-periodic structure in annD embedding space,V , by the physical subspace, whereV is the directsum ofVE andVI (V = VE + VI ) andVI is the orthogonal complement (the perpendicularspace) ofVE . The symmetry groups for the 3D lattice-periodic systems have been knownfor a long time (Fedorov 1885) and their generalizations to higher-dimensional spaces havebeen studied by a number of scientists. At present, all the point groups have been discussedfor n = 4–9 (Ryskov 1972, Plesken and Pohst 1977, 1980, Brownet al 1978). Owing to thedistinguished physical subspace, of course, the crystallography of a QS is not strictly the same asin nD crystallography. This means that the symmetry groups describing QS arenD symmetrygroups satisfying appropriate additional requirements (Janssen 1991, 1992, Wijnands andThiers 1992, Wijnands and Janssen 1993). The space groups have been investigated forpolygonal and icosahedral symmetries (Mackay 1985, Gahler and Rhyner 1986, Gahler 1990,Janssen 1986, Rokhsaret al 1988, Levitov and Rhyner 1988, Rabsonet al 1991, Lifshitz andMermin 1994). A systematic classification of symmetry groups is given explicitly by Wanget al (1997) for 1D QCs and by Huet al (1996b) for 2D QCs. A detailed account of symmetrydetermination and structure analysis can be found in the review article by Yamamoto (1996).
In this section we follow Janssen, but introduce a slightly more transparent version than theone originally put forward for the construction of the symmetry operations for a QS. ConsiderannD lattice6 in V with its reciprocal lattice6∗. A crystallographic operationR, that leaves6 invariant, acts on a basisbi (i = 1, 2, . . . , n) according to
Rbi =n∑j=1
[0(R)]jibj (i = 1, 2, . . . , n) (2.1)
where0(R) is the representation matrix of the symmetry operation,R, in this basis. It isintegral and of finite-orderN , which is the smallest number such that [0(R)]N = E (E isthe unit matrix). Therefore, the eigenvalues of the matrix0(R) are allN th unit roots. Itscharacteristic polynomial
f (λ) = det[0(R)− λE] (2.2)
is a polynomial with integral coefficients, which has a unique decomposition into prime factors:
f (λ) =∏µ
fµ(λ) (2.3)
where each factorfµ(λ) is a cyclotomic polynomial, all the roots of which have the formλ = ρp (ρmµ = 1,p is an integer between 0 andmµ and is a prime number relative tomµ). λis a primitivemµth root of unity andmµ is a divisor ofN (mµ/N ). The number of such roots,nµ, is determined by the Euler function (van der Waerden 1955)
nµ = E(mµ). (2.4)
If mµ can be written in the form
mµ =l∏i=1
pνii (2.5)
Physical properties of quasicrystals 5
wherepi (i = 1, 2, . . . , l) are different primes andνi (i = 1, 2, . . . , l) are natural numbers,then
E(mµ) = mµl∏i=1
(1− 1
pi
). (2.6)
The basis forV can be chosen in such a way as to decomposeV into a direct sum of someinvariant subspacesVµ. Each of them is irreducible over the integral ring. For such a choice ofthe basis,0(R) is block diagonalized. The characteristic polynomial of each submatrix is thecyclotomic polynomialfµ(λ) and each invariant subspaceVµ is spanned by the eigenvectorsthat belong to the eigenvalues, i.e. the roots offµ(λ). Following the notation used by Hermann(1949),R can now be presented as
R ∼ {m1, m2, . . .}∑µ
E(mµ) = n. (2.7)
For definiteness, the order of the integersmµ in this notation is arranged in such a waythat the integer corresponding to the periodic dimension precedes those corresponding tothe quasiperiodic dimension and the integer corresponding to 3D physical spaceVE precedesthose corresponding to the (n − 3)D perpendicular spaceVI . As an example, forn = 5 thesymmetry operationR = {12222} has five invariants. One 1D eigenspace is spanned by theeigenvector belonging to the eigenvalue of 1 (the first root of unity), The remaining space isspanned by four eigenvectors belonging to the eigenvalue of−1 (the second root of unity).All invariant subspaces are 1D.R = {15} has two invariant subspaces. One is an eigenspacecorresponding to 1 and the other is 4D eigenspace corresponding to the fifth primitive root ofunity.
Because QSs are the projection of periodic structures onto the physical space, one mayconsiderR as the symmetry operation of QCs. However, forR to be such a symmetry operationsome additional appropriate conditions must be satisfied.
(1) Consider a crystallographic operationR = {mµ} acting onV , which can be decomposedinto a direct sum ofVµ. Each of them is irreducible over the integral ring, but may bereducible over the real field. This is because the cyclotomic polynomial is solvable overthe complex field. Moreover, the complex roots occur in pairs, i.e., ifλ is a solution offµ(λ) = 0, the complex conjugateλ∗ is also a solution. Therefore, over the real numbers,Vµ can be decomposed into a sum of subspaces, the dimension of which is one for thecase of real roots or two for the case of complex roots. For example, over the integral ringthe correspondingVµ is a 4D irreducible space underR = {5}. Admitting real numbers,Vµ can be decomposed into twoR-irreducible subspacesV1 andV2, where they are both2D. One belongs toρ andρ−1, and the other belongs toρ2 andρ−2 whereρ, ρ−1, ρ2
andρ−2 are the fifth primitive roots of unity (ρ5 = 1). If the physical spaceVE containsone fully, sayV1, but has a 1D intersection withV2, a non-zero vector in this intersectionobtains a component in the perpendicular spaceVI after the transformationR. ThusR isnot a symmetry operation. Such a transformation is called a mixing transformation and asymmetry operation used for the description of QS must be non-mixing.
(2) In order to allow a QS a non-mixing transformation,R = {mµ}must satisfy the followingconditions:
• Either VI carries full spaceVµ and every correspondingmµ also occurs in thedecomposition ofVE ,• or, if a subspaceVµ has a non-zero intersection with bothVE andVI , thenVE contains
a 2DR-irreducible subspace ofVµ andVI contains the remaining (mµ−2)D subspace.
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First, if there exists a subspaceVµ for whichmµ does not occur for subspaces ofVE ,then since the projection ofVµ on VE is an empty set, this subspace cannot contributeto incommensurability. This means that such anR cannot produce a QS, as is the casefor R = {1322}. Second, the subspaceVµ carries an integral representation ofR whichcannot be reduced by rational matrices. IfVµ has a non-vanishing intersection with bothVE andVI , basis vectors forVµ cannot belong toVE and the structure inVE is not latticeperiodic. An example isR = {15}. mµ = 5 characterizes a 4D irreducible spaceVµover the integral ring. As stated above,Vµ can be decomposed into twoR-irreduciblesubspaces. If one belongs toVE and the other toVI , the structure inVE is certainly notlattice periodic.
Conditions (1) and (2) must be satisfied by annD crystallographic operation in order toallow a QS. Once the symmetry operations are determined, we can construct the symmetrygroups of QCs by group definition. In the following section we discuss them.
2.2. Symmetry groups
For simplicity we restrict our discussion to the point groups of QCs withr = n (r is the rank ofthe Fourier module). A 1D QC can be obtained by projecting a 4D embedding space onto thethree-dimensional (3D) physical space. In this case the 12 possible symmetry operations are1111, 2211, 311, 411, 611, 2222, 1122, 622, 422, 322, 1222 and 2111 (Janssen 1992). In otherwords, they are then andn axes (n = 1, 2, 3, 4, 6) along thez direction which is perpendicularto thexy periodic plane, and also the twofold axes 2h (horizontal twofold rotation) and2h = mv(vertical mirror reflection). Moreover, sincen = 1 · n (n = 2, 3, 4, 6), 2h = mv · 1, in fact,there are seven basic symmetry operations:n-fold rotations (n = 1, 2, 3, 4, 6), inversion1and vertical mirror reflectionm = mv. All the point groups of 1D QCs can be generated bythese seven symmetry operations. First, the first-type operations (1, 2, 3, 4, 6, m) generate thefollowing ten point groups:
1, 2, 3, 4, 6, m, 2mm, 3m, 4mm, 6mm. (2.8)
Next, if any groupH contains a second-type operation (a second-type operation is a compoundoperation which is a product of inversion and one of the first-type operations), thenH certainlycontains a subgroupH1 of index two which consists of all the first-type operations because theproduct of any two second-type operations is a first-type one, i.e.
H = H1 + 1 · aH1 (2.9)
wherea is a first-type operation. Let1 · a = 1 andH1 be any member given in equation (2.8).Then we have the following ten point groups:
1, 2/mh, 3, 4/mh, 6/mh, 2h/m,mhmm, 3m, 4/mhmm, 6/mhmm.
(2.10)
If we now choose not 1 but 2, 3, 4, 6 ormas the representative elementa, then1·a = mh,3,4,6or 2. Notice thatH = H1 + (1 · a)H1 is isomorphic toH ′ = H1 + aH1. We can derive otherpoint groups for 1D QCs by finding subgroupsH1 of index two from the ten point groupsH ′
given in equation (2.8) and then changing the operationa into the corresponding1 · a. SuchH are listed as follows:
H ′ 2, 4, 6, m, 2mm, 3m, 4mm, 6mm
H1 1, 2, 3, 1, 2, m, 3, 4, 2mm, 6, 3m (2.11)
a 2, 4, 6, m, m,2, m, m,4, m,6.
Physical properties of quasicrystals 7
Table 1. Systems, Laue classes, point groups and the number of independent elastic constants for1D QCs. NC,NK andNR are the numbers of independent elastic constants associated with thephonon field, the phason field and the phonon–phason coupling, respectively.
Systems No of Laue classes Point groups NC NK NR Sum
Triclinic 1 1, 1 21 6 18 45Monoclinic 2 mh, 2/mh 13 4 8 25
3 2h,m, 2h/m 13 4 10 27Orthorhombic 4 2h2h2,mm2, 2hmmh,mmmh 9 3 5 17Tetragonal 5 4,4, 4/mh 7 2 4 13
6 42h2h, 4mm, 42hm, 4/mhmm 6 2 3 11Trigonal 7 3,3 7 2 6 15
8 32h, 3m,3m 6 2 4 12
Hexagonal 9 6,6, 6/mh 5 2 4 11
10 62h2h, 6mm, 6m2h, 6/mhmm 5 2 3 10
Replacinga by 1 · a, we obtain the other 11 point groups:
mh, 4, 6, 2h, 22h2h, 2hmmh,32h, 42h2h, 42m, 62h2h, 62m.
(2.12)
These 31 point groups of 1D QCs may also be derived by the following considerations. Thepossible operationsn, n (n = 1, 2, 3, 4, 6), 2h and2h = m all belong to crystallographic pointoperations, hence any point group of 1D QCs must be one of the 32 crystallographic pointgroups. By excluding five cubic point groups 23,m3, 432,43m, andm3m which containoblique axes and considering 2 and 2h, m andmh, 2/mh and 2h/m, 2mm and 2hmmh asdifferent point groups, we again obtain 31 point groups of 1D QCs. These 31 groups can bedivided into six systems and ten Laue classes. All are given in table 1.
In a similar manner, we can find all possible point groups of 2D QCs and 3D QCs. Forexample, one consider a 2D QC with the point symmetryK = 5 (C5) generated by a fivefoldrotation. This generator can be represented by
0(R) =
1 0 0 0 00 0 0 0 −10 1 0 0 −10 0 1 0 −10 0 0 1 −1
(2.13)
which is of type{15}. This means that the symmetry operation{15} can generate the pointgroup 5. Again, the point groupK2 = 5m (C5v) has the representation with
0(R1) =
1 0 0 0 00 0 0 0 −10 1 0 0 −10 0 1 0 −10 0 0 1 −1
0(R2) =
1 0 0 0 00 0 0 0 10 0 0 1 00 0 1 0 00 1 0 0 0
. (2.14)
The fivefold rotationR1 and the mirrorR2 are the generators ofK2. The former is of type{15}and the latter is of type{11212}. It follows that the symmetry operations{15} and{11212}can generate the point group 5m. Similarly, the symmetry operations{15} and{21212} cangenerate the point group 52. The other point groups for 2D QCs can be constructed in the samefashion. It should be noted, however, that a QC is not necessary to be associated with non-crystallographic point-group symmetry (Janssen 1992). Thus it is possible for crystallographicsymmetry groups to allow QS. This is the case for the cubic symmetry (Fenget al 1989).
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Table 2. Systems, Laue classes, point groups and the number of independent elastic constants for2D QCs.
Systems No of Laue classes Point groups NC NK NR Sum
Triclinic 1 1, 1 21 21 36 78Monoclinic 2 2,m, 2/m 13 13 20 46
3 12, 1m, 12/m 13 12 18 43Orthorhombic 4 2mm, 222,mmm,mm2 9 8 10 27Tetragonal 5 4,4, 4/m 7 7 10 24
6 4mm, 422,4m2, 4/mmm 6 5 5 16Trigonal 7 3,3 7 7 12 26
8 3m, 32,3m 6 5 6 17
Hexagonal 9 6,6, 6/m 5 5 8 18
10 6mm, 622,6m2, 6/mmm 5 4 4 13
Pentagonal 11 5,5 5 5 6 16
12 5m, 52,5m 5 4 3 12Decagonal 13 10,10, 10/m 5 3 2 10
14 10mm, 1022,10m2, 10/mmm 5 3 1 9Octagonal 15 8,8, 8/m 5 5 2 12
16 8mm, 822,8m2, 8/mmm 5 4 1 10Dodecagonal 17 12,12, 12/m 5 5 0 10
18 12mm, 1222,12m2, 12/mmm 5 4 0 9
Additionally, since a twofold rotation and a mirror may be along the quasiperiodic plane or theperiodic axis, there are six point groups (2, 12,m, 1m, 2/m and 12/m) in the monoclinic caseand four point groups (222, 2mm, mm2 and 2/mmm) in the orthorhombic case, in contrastwith conventional crystals which have only three point groups for each system, monoclinic ororthorhombic. Table 2 lists all possible point groups of 2D QCs with Fourier modulus of rank5. There are 57 point groups divided into ten systems and 18 Laue classes.
3D QCs have 60 point groups. They are: (1) 32 crystallographic point groups consistingof only crystallographically allowable symmetry operations and (2) 28 non-crystallographicpoint groups containing crystallographically forbidden operations, i.e. icosahedral point groups(235,m35) and 26 point groups with 5-, 8-, 10- and 12-fold symmetries (5,5, 52,5m, 5m andN ,N ,N/m,N22,Nmm,Nm2,N/mmm,N = 8, 10, 12).
2.3. Experimental determination of symmetry groups of QCs
Very soon after Shechtman’s discovery, many experimental techniques commonly used inmaterial science were employed for the structure determination of QCs. Among themdiffraction techniques are the methods of choice. Convergent-beam electron diffraction(CBED) allows experimental determination of symmetry groups from very small crystallineareas of QCs (Terauchiet al 1994, Tanakaet al 1994). Single-crystal x-ray diffraction andneutron diffraction are also very powerful and well established tools for quantitative structuredetermination (Steurer 1989, 1994, Gratiaset al1988). Along with the diffraction techniques,high-resolution electron microscopy and various after spectroscopic techniques have alsoproved to be useful methods (Li and Liu 1986, Hiragaet al 1987, Koopmanset al 1987,Leeet al 1988). An exhaustive scan of all the methods employed is beyond the scope of thisreview. We limit ourselves to the presentation of some experiments which reveal the majorfeatures of the quasicrystalline structures.
After the experimental observation of the first quasicrystalline phase a considerable
Physical properties of quasicrystals 9
number of publications were devoted to symmetry and structure determination. For 3D QCs,such as i–Al–Mn and i–Al–Mn–Si, their point groups have been detemined to bem35 andtheir space groupsPm35 (Shechtmanet al 1984, Bancelet al 1985, Chen and Chen 1986,Mukhopadhyayet al 1987, Tanakaet al 1987, Thangarajet al 1987, Yamamoto and Hiraga1988, Cahnet al 1988, Gratias 1988, Janotet al 1989). Further investigation revealed thatfor the systems Al–Cu–Me (Me= Fe, Ru, Os) and Al–Me–Pd (Me= Mn, Re) the spacegroups areFm35 (Steurer 1994). The octagonal phases were detected in the Cr–Ni–Si andMn–Si systems (Wanget al 1987, Caoet al 1988). The point groups are 8/m or 8/mmm.The first dodecagonal phase was found in the Ni–Cr system and structural analyses showedthe point group to be 12/mmm (Ishimasaet al 1985, Chenet al 1988). The decagonalphases were observed in several systems, such as Al–Mn, Al–Fe and Al–Mn–Fe (Bendersky1985, 1987, Chattopadhyayet al 1985, Funget al 1986, Koopmanset al 1987) and the pointgroups were found to be 10/mmm, and the space groupsP105/mmc for Al–Mn and Al–Fe(Kumaret al 1986, Bridgeset al 1987, Yamamoto and Ishihara 1988, Steurer 1989, Steurerand Mayer 1989). Later, Steurer and Kuo (1990) used x-ray structure analysis of an annealedAl65Cu20Co15 alloy. Hiragaet al (1991) took high-resolution electron microscopy (HREM)images of an Al70Ni15Co15 alloy at an accelerating voltage of 400 V. Steureret al (1993)performed x-ray structure analysis of an Al70Ni15Co15 alloy. All this showed that the spacegroups areP105/mmc. Recently, Tanakaet al (1993), Saitohet al (1996) and Tsudaet al(1996), using the CBED and HREM methods, revealed temperature and composition changesof the space groups for some decagonal QCs. The space group of the melt-quenched Al–Cu–Co alloys transforms from non-centrosymmetricP10m2 to centrosymmetricP105/mmc atabout 600◦C. The space group of Al70NixFe30−x (10< x < 17) isP10m2, whereas the spacegroup of Al70NixFe30−x (17< x < 20) isP105/mmc.
Even though experimental techniques are developed and the collection of experimentaldata on QC structure studies has grown, the results of last 14 years’ symmetry and structuredetermination are still incomplete. Further steps in this direction have to be taken.
2.4. Symmetry breaking and structural transitions
Immediately after the discovery of QCs, the imperfections in QCs were also noted. Thestriking characteristic of QCs is the existence of sharp Bragg peaks. However, experimentsindicated peak broadening and subtle distortion of the diffraction patterns (Goldmanet al1990). These observations have revealed some deviations of real alloys from the ideal QCmodel. Another interesting feature is that many alloys exhibiting a quasicrystalline phase havecrystalline phases of similar composition and local structure. The related crystalline phasesoften coexist with the quasicrystalline phase (Audier and Guyot 1986, Sainfort and Dubost1986). These crystalline phases are called approximants because they can be recovered bysubstituting rational ratios existing in such approximants for the irrational one of the QC (Elserand Henley 1985, Henley and Elser 1986). All these findings show the possibility of structuraltransitions in QCs (Audier and Guyot 1989, 1990). Some authors suggested that such structuraltransitions are mainly induced by the phason strain field. Lubenskyet al (1985, 1986) andHornet al (1986) discussed how the phason strain can lead to peak broadenning and distortionin QC diffraction patterns. Bancel (1989, 1990), Widom (1990), Henley (1990), Ishii (1989,1992) and Hatwalne and Ramaswamy (1990) analysed the structural instability and transitiondriven by phason mode locking and softening in QCs. Jeong and Steinhardt (1993) appliedthe phason dynamics of an energetically stabilized tiling model for the decagonal phason todemonstrate the transition from a locked phase to unlocked phase at finite temperature.
Structural transitions can be analysed in the framework of Landau theory (Landau and
10 C Z Hu et al
Lifshitz 1968). In what follows, we show how to develop a Landau theory for the phasetransition in QCs connected with symmetry change and derive the possible ways of symmetrychange with the group theoretical method. According to Landau theory the density of theordered, low-temperature phase is taken to be of the form
ρ(r) =∑
ρGeiG·r (2.15)
where
ρG = |ρG|eiφG. (2.16)
Moreover, we can parametrize the independent component ofφG with two displacement fields,the phonon fieldu and the phason fieldw. Both are functions of the position vector in thephysical space only (Bak 1985).
φG = G · u +G′ ·w (2.17)
whereG is the reciprocal vector in the physical space andG′ is the conjugate vector in theperpendicular space. To the approximation we are interested in it is sufficient to consider onlythe linear effect during the phase fluctuation ofρG. Since a linear phonon strain does notchange the configuration of the unit cell, it is plausible to ascribe the structural transitions inQCs to the phason strain field. In this case
ui(r) = ui(r0)
wi(r) = wi(r0) +Wij4rj(2.18)
wherer0 is the quiescent distance in the absence of the strains and
Wij = ∂jwi. (2.19)
By inserting equation (2.18) into (2.15), we find
ρ = ρ(r) =∑
ρGeiG·r0eCijWij = ρ0eCijWij (2.20)
whereρ0 = ρ(r0) andCij (∝ 4rj ) are infinitesimal parameters. SinceCij becomes arbitrarilysmall near the transition point, equation (2.20) can be written as
ρ = ρ0(1 +CijWij ) = ρ0 + δρ. (2.21)
For a phase transition connected with a change in the structure of the lattice,ρ0 has the symmetrygroupK0, ρ and henceδρ have the symmetryK, which is a proper subgroup ofK0. In otherwords,ρ0 is invariant with respect to all elements of the groupK0, i.e. it forms the identityrepresentation ofK0, andδρ forms the identity representation of the subgroupK. This meansthat all the coefficientsCij should vanish at the trasition point except those corresponding to thebasis of the identity representation ofK. Therefore, the problem of determining the possibletypes of symmetry change in a phase transition is reduced to finding the non-vanishing phasonstrains, which form a basis of the identity representation of the symmetry groupK. Usingthe reduction of a representation of the group with its subgroup, we can find the invariantcomponents and then determine any non-vanishing phason strains (the locking of a particularmodel of phason). As an example, consider the octagonal groupD8 (822), which has sevenirreducile representations (see table 3). The maximal crystallographic subgroup ofD8 isD4.The phason strains∂jwi transform under
05× 07 = 03 + 04 + 06. (2.22)
By reducing these irreducible representations ofD8 with respect toD4, one can find that only1D representation03 contains the identity representation ofD4. This means that03 mode is alocked-in phason mode and the component of this mode is a non-vanishing phason strain, whichcauses the symmetry changeD8→ D4. In this way, Huet al (1996a) analysed the symmetry
Physical properties of quasicrystals 11
Table 3. Characters ofC8v symmetry.
C8v(8mm) ε α α2 α3 α4 β αβ
02 1 1 1 1 1 −1 −103 1 −1 1 −1 1 1 −104 1 −1 1 −1 1 −1 1
05 2√
2 0 −√2 −2 0 006 2 0 −2 0 2 0 0
07 2 −√2 0√
2 −2 0 0
changes:D8 → D4, D10→ D2, D12→ D6 andD12→ D4, and calculated non-vanishingphason strains for all the possible symmetry breaking and structural transitions in octagonal,decagonal and dodecagonal QCs with the help of group theory. Such symmetry breakingphase transitions are observed experimentally (Maiet al1989, Zhang and Kuo 1990, Krumeichet al 1994). The results can account for these experiments. Ishii (1989, 1992) discussed thesymmetry breaking and structural transitions in icosahedral QCs. Three possible ways aregiven to explain these symmetry changes:Y → T , Y → D5 andY → D3. Non-vanishingphason strains are also derived for these symmetry changes.
There is an alternative approach to phase transitions of QCs based on the random tilingmodel (Elser 1985). With the help of this model an analytical (not explicitly group theoretical)treatment is introduced (Widomet al1989, Oxborrow and Henley 1993). Combining a Landautheory of interacting incommensurate phases and numerical transfer matrix calculations, Liet al (1992) investigated phase transitonsD8→ D4/D2. Applying the Bethe-ansatz techniqueto solve the square–triangle tiling model, Widom (1993) and Kalugin (1994) found an exactlysolvable model displaying phase transitionsD12→ D4/D6.
3. Physical property tensors
3.1. Thermodynamics of equilibrium properties
An important question that has existed since the first discovery of QCs concerns theirthermodynamical stability (Bihamet al 1986, Schaefer and Bendersky 1988, Widomet al1989). Thermodynamically stable icosahedral alloys with a high degree of quasicrystallinitywere discovered by Tsaiet al (1987) in AlCuMe (Me= V, Cr, Fe, Os) ternary alloys. Later,a new stable quasicrystalline phase in AlMnPd and AlPdRe alloys was also reported by Tsaiet al (1990). Thus, it is possible to consider a QC as a thermodynamic system and to studythe thermodynamics of its equilibrium properties (Yanget al 1996a, Huet al 1997b). As weknow, a thermodynamic system shows various macroscopic effects when various influences areapplied. The physical properties of a QC describes its response to externally applied influencessuch as forces and fields. It should be noted that there are two kinds of displacement fields inQCs: the phonon and phason displacementsu andw. Consequently, there are two kinds ofstrain fields: the phonon strainEij = 1
2(∂jui + ∂iuj ) and the phason strainWij = ∂jwi , andtwo kinds of stress fields: the phonon stressTij and the phason stressHij (Lubensky 1988,Ishii 1989, Dinget al 1993). Obviously, these characteristics will have an influence on thethermodynamics of QCs. Below, we restrict the thermodynamic system to a closed systemwhich does not interchange matter with its surrounding. Moreover, we assume that energy isonly transferred to the system by the mechanical work done on the system by surface tractionand body forces, by heat transfer through the boundary, and by the electrical and magnetic
12 C Z Hu et al
work done on the system. For the reversible processes the fundamental thermodynamic lawgives
dU = Tij dEij +Hij dWij + Fi dDi + θ dS (3.1)
where dU is the increase in the internal energy;Tij dEij andHij dWij are the work per unitvolume done by the stressesTij andHij in changing the strains dEij and dWij , respectively;Fi dDi is the electric or magnetic work done on the system withFi being the electric ormagnetic field intensity vector andDi the electric displacement or magnetic induction vector;andθ dS is the heat flow withθ andS being the absolute temperature and entropy, respectively.One can define some other thermodynamic functions, such as the Gibbs free energy
G = U − TijEij −HijWij − FiDi − θS (3.2)
and the Helmholtz free energy
9 = U − θS. (3.3)
If we differentiate equation (3.2) and use (3.1) to substitute dU , we then obtain
dG = −Eij dTij −Wij dHij −Di dFi − S dθ. (3.4)
We may considerG to be a function ofTij ,Hij , Fi andθ and write
dG =(∂G
∂Tij
)H,F,θ
dTij +
(∂G
∂Hij
)T ,F,θ
dHij +
(∂G
∂Fi
)T ,H,θ
dFi +
(∂G
∂θ
)T ,H,F
dθ. (3.5)
Comparing with equation (3.4), we have(∂G
∂Tij
)H,F,θ
= −Eij(∂G
∂Hij
)T ,F,θ
= −Wij(∂G
∂Fi
)T ,H,θ
= −Di
(∂G
∂θ
)T ,H,F
= −S.(3.6)
Let us choose (Tij ,Hij , Fi , θ ) as a set of independent variables and (Eij ,Wij ,Di , S) as a set ofdependent variables. Further, the differentials of (Eij , Wij , Di , S) can be represented as onematrix equation:
dEijdWij
dDi
dS
=
(∂Eij∂Tkl
)H,F,θ
(∂Eij∂Hkl
)T ,F,θ
(∂Eij∂Fk
)T ,H,θ
(∂Eij∂θ
)T ,H,F(
∂Wij
∂Tkl
)H,F,θ
(∂Wij
∂Hkl
)T ,F,θ
(∂Wij
∂Fk
)T ,H,θ
(∂Wij
∂θ
)T ,H,F(
∂Di∂Tkl
)H,F,θ
(∂Di∂Hkl
)T ,F,θ
(∂Di∂Fk
)T ,H,θ
(∂Di∂θ
)T ,H,F(
∂S∂Tkl
)H,F,θ
(∂S∂Hkl
)T ,F,θ
(∂S∂Fk
)T ,H,θ
(∂S∂θ
)T ,H,F
dTkldHkldFkdθ
. (3.7)
Similarly, from equations (3.3) and (3.1) it follows that
d9 = Tij dEij +Hij dWij + Fi dDi − S dθ. (3.8)
In the case for which (Eij ,Wij ,Di , θ ) are a set of independent variables we have:
dTijdHijdFidS
=
(∂Tij∂Ekl
)W,D,θ
(∂Tij∂Wkl
)E,D,θ
(∂Tij∂Dk
)E,W,θ
(∂Tij∂θ
)E,W,D(
∂Hij∂Ekl
)W,D,θ
(∂Hij∂Wkl
)E,D,θ
(∂Hij∂Dk
)E,W,θ
(∂Hij∂θ
)E,W,D(
∂Fi∂Ekl
)W,D,θ
(∂Fi∂Wkl
)E,D,θ
(∂Fi∂Dk
)E,W,θ
(∂Fi∂θ
)E,W,D(
∂S∂Ekl
)W,D,θ
(∂S∂Wkl
)E,D,θ
(∂S∂Dk
)E,W,θ
(∂S∂θ
)E,W,D
dEkldWkl
dDk
dθ
. (3.9)
In formula (3.7) or (3.9) there are 9 +(n − 3) × 3 + 3 + 1equations for a QC of rankn. Forexample,n = 6 for icosahedral and cubic QCs, andn = 5 for pentagonal, octagonal, decagonal
Physical properties of quasicrystals 13
and dodecagonal QCs. Each of the matrix elements is a measure of a physical effect. Theseeffects can be divided into two kinds: one consists of principal effects represented by the partialderivatives on the leading diagonal of the matrix, while the other consists of coupled effectsrepresented by the off-diagonal elements. Principal effects include elasticity, permitticity(permeability) and heat capability. Coupled effects include piezoelectricity, pyroelectricityand thermoelastic effects. Each effect is associated with a matter constant that characterizesthe response of a QC to an applied force or field. Some points are worth noting.
(1) Such a matter constant is usually labelled a tensor of rankn according to its behaviourunder a coordinate transformation. For example, the heat capacity is a scalar (zero-ranktensor), the pyroelectric coefficient is a vector (first-rank tensor), the thermal expansioncoefficient is a second-rank tensor, the piezoelectric constant is a third-rank tensor and theelastic constant is a fourth-rank tensor. These tensors describing the physical propertiesof a material are called physical property tensors.
(2) If we differentiate the first two relations of equation (3.6) with respect toFk and the thirdrelation with respect toTij andWij , we can obtain the following relationship betweencoefficients of direct and converse piezoelectric effects:
−(
∂2G
∂Tij ∂Fk
)θ
=(∂Eij
∂Fk
)θ
=(∂Dk
∂Tij
)θ
= d(1)kij = d(1)ijk
−(
∂2G
∂Hij ∂Fk
)θ
=(∂Wij
∂Fk
)θ
=(∂Dk
∂Hij
)θ
= d(2)kij = d(2)ijk.(3.10)
Similar relationships hold also for other direct and converse effects.(3) Permittivity, permeability, heat capacity and pyroelectricity take the same forms as in
conventional crystals, while other properties relevant to mechanical quantities containadditional terms related to phason strainWij or stressHij in the perpendicular subspace.
3.2. Determination of non-vanishing independent components
The macroscopic physical properties of a QC are invariant under all symmetry operationswhich the QC possesses. There then ought to exist some combinations of the componentsof a physical property tensor that remain constant under the symmetry operations. Thesecombinations form a set of invariants of the tensor. Only the tensor components appearing inthe invariants are non-vanishing. Moreover, the number of independent tensor components isequal to the number of invariants. The number and form of invariants and hence a tensor canbe determined by the group representation theory. For ordinary crystals most common tensorshave already been given in many texts (Bhagavantam, 1966). In principle, it is straightforwardto generalize the method for determing a physical property tensor from ordinary crystals toQCs. However, according to the higher-dimensional description of QCs, a QC structure isobtained by projecting a higher-dimensional lattice (V ) onto the physical space (VE) whereV = VE+VI withVI being the perpendicular space. A vector inVE transforms under the vectorrepresentation (0A) of the symmetry group of the structure considered, whereas a vector inVItransforms under another irreducible representation (0B). As a result, the practical calculationusually becomes tedious and complicated. To render this problem more tractable we first focuson the transformation matrix and its reduced form of the strain (stress) field (Juretschke 1974,Yanget al 1995b).
Let us consider a 2D QC with the point symmetry groupCn. If A andB are the coordinatetransformation matrices of the physical and perpendicular subspaces, respectively, then
A =[M(α) 0
0 1
]B = M(β) (3.11)
14 C Z Hu et al
for a rotation through an angleα = 2π/n about thez-axis in the physical space where
M(θ) =[
cos(θ) − sin(θ)sin(θ) cos(θ)
](3.12)
α = 2π/n, n = 5, 8, 10, 12 andβ = pα, p = 3, 3, 3, 5 for the case of QCs with non-crystallographic symmetries (Yanget al 1996b). Since the generators of a group are thoseelements that by repeated application among themselves generate the whole group, it is notnecessary to consider all operations, but only the generators. The point groupN (Cn) has onegenerator, i.e., ann-fold rotationα = 2π
n. For a givenα the transformation of the phonon
strain field is
(∂ × u)′ ={[M(α) 0
0 1
]×[M(α) 0
0 1
]}(∂ × u) (3.13)
whereas the transformation of the phason strain field is
(∂ × w)′ ={[M(α) 0
0 1
]×M(β)
}(∂ × w) (3.14)
where× labels a direct product. The above transformation matrices of the phonon and phasonstrain fields can be reduced through a similarity transformation into block diagonal form:[
E11′ − E22
′
2E12′
]= M(2α)
[E11− E22
2E12
][E13′
E23′
]= M(α)
[E13
E23
]E′11 +E22
′ = E11 +E22
E′33 = E33[W11
′ +W22′
W21′ −W12
′
]= M(β − α)
[W11 +W22
W21−W12
][W11
′ −W22′
W21′ +W12
′
]= M(β + α)
[W11−W22
W21 +W12
][W13
′
W23′
]= M(β)
[W13
W23
]
(3.15)
where the antisymmetric part of the phonon strain field is associated with a rotation of the solidwithout distortion, which we can eliminate.
These seven relations in equation (3.15) characterize the transformation properties of thephonon and phason strain fields under the symmetry group of a QC. Using these relations, wecan easily construct all invariant linear combinations of strain tensor components, and thenobtain the elastic constant and other matter constants related to the strain field. In the generalcase a symmetry group may have more than one generator, such as inversion and reflection.The extension of the mathematical treatment to include several generators is direct. In thiscase one should consider not one generator, but all generators. First, we detemine the numberof independent tensor components, and then illustrate the method for constructing the explicitform of the tensors in sections 3.3 and 3.4.
It is easy to see that the character analysis offers an elegant method for determining thenumber of independent tensor components without having to construct the explicit form of thetensor invariants. In fact, the number of independent tensor components is equal to the numberof times that the identity representation is contained in this tensor representation, i.e.
n = 1
|G|∑g∈G
χ(g) (3.16)
Physical properties of quasicrystals 15
where|G| is the order of a given groupG, g an element inG andχ the character of the tensorrepresentation ofG. In order to determine the value ofn in equation (3.16), Yanget al (1994)have given the formulae for calculating the representation matrices0(g) and the correspondingcharactersχ(g) for various types of tensors (see appendix A).
Once the symmetry group of a QC is given, the representation matrices and correspondingcharacters of a physical property tensor can be derived, and then the number of independentcomponents of the tensor can be readily obtained from equation (3.16). However, only thoseproperties related to mechanical quantities must be inferred from considerations of both phononand phason fields. We shall restrict our discussion to those physical property tensors, whichinclude thermal expansion coefficientsα(1)ij , α(2)ij , piezoelectric coefficientsd(1)ijk , d
(2)ijk (where
superscripts (1) and (2) denote the components associated with the phonon and phason fields)and elastic constantsCijkl , Kijkl , Rijkl . As an example, we consider octagonal QCs withC8v (8mm) symmetry. This point group has 16 elements, seven conjugacy classes and sevenirreducible representaions (cf table 3), four of which are 1D and three 2D. Two generators arethe eightfold rotationα and the mirror operationβ. The action ofα andβ on the basis vectorsis given (Steurer 1990, Huet al 1993a, b) by
0(α) =
0 1 0 0 00 0 1 0 00 0 0 1 0−1 0 0 0 00 0 0 0 1
0(β) =
1 0 0 0 00 0 0 −1 00 0 −1 0 00 −1 0 0 00 0 0 0 1
. (3.17)
The matrix representation0 is reducible. The reduction is
0 = 05 + 01 + 07. (3.18)
It follows that a vector inVE transforms under05 + 01, whereas a vector inVI transformsunder07. In this caseA = 05 +01 andB = 07. Using the character analysis we can find (seethe appendix):
(1) the number of independent components is two forα(1)ij and zero forα(2)ij , respectively;
(2) the number of independent components is three ford(1)ijk and zero ford(2)ijk , respectively;
(3) the numbernC , nK , andnR of independent elastic constantsCijkl ,Kijkl andRijkl are
nC = 5, nK = 4, nR = 1 (3.19)
whereCijkl ,Kijkl andRijkl denote the elastic constants associated with the phonon field,the phason field and the phonon–phason coupling, respectively.
The numbers of the independent tensor components for other QC symmetries can easily becalculated by the method demonstrated in the appendix. All the results are listed in tables 4–6as subscripts attached to the corresponding physical property tensors expressed in matrix form.
3.3. Elastic constant tensors
Over the years since their discovery the physical properties expected of QCs have been thesubject of theoretical investigations. A large number of contributions deal with the elasticity.Unlike periodic crystals, QSs have a degree of freedom associated with relative translation ofincommensurate sublattices or, equivalently, with special types of atomic rearrangements. Theelementary excitations arising from this degree of freedom are called phasons and parametrizedby a phason variable. Therefore, the elastic energy of a QC depends on both phonon and phasonfields. The continuum elastic-hydrodynamic theory has been developed for QCs (Lubenskyet al 1985, 1986, Bak 1985a, b, Kaluginet al 1985a). Levineet al (1985), Bak (1985b) and
16 C Z Hu et al
Table 4. Elastic constants of 2D QCs with noncrystalline symmetries. All 2D QCs withnoncrystalline symmetries have the same phonon elastic constant (Cijkl listed below in (a):
whereC66 = C11−C122 . For Laue class 11 (the number of Laue classes is the same as in
table 2) the phason fieldKijkl results are presented in (b) and the phonon–phason couplingRijkl results in (c). For Laue class 12 all the results are the same as Laue class 11 except forK6 = R2 = R3 = R5 = 0(z ‖ x1, m ⊥ x1), andK7 = R2 = R4 = R6 = 0(z ‖ x2, m ⊥ x2).For Laue class 13,K6 = K7 = R3 = R4 = R5 = R6 = 0 and for Laue class 14,K6 = K7 = R2 = R3 = R4 = R5 = R6 = 0. For Laue class 15 the phason fieldKijklresults are presented in (d), where6 = K1 + K2 + K3, and the phonon–phason couplingRijklresults are presented in (e). For Laue class 16,K5 = R2 = 0. For Laue class 17, the same phasonelastic constant as Laue class 15 is found, but no constants associated with the phonon–phasoncoupling. Laue class 18 shows the same phason elastic constant as Laue class 16, but there are noconstants associated with the phonon–phason coupling.(a)
11 22 33 23 31 12
11 C11 C12 C13 0 0 022 C12 C11 C13 0 0 033 C13 C13 C33 0 0 023 0 0 0 C44 0 031 0 0 0 0 C44 012 0 0 0 0 0 C66
(b)
11 22 23 12 13 21
11 K1 K2 K7 0 K6 022 K2 K1 K7 0 K6 023 K7 K7 K4 K6 0 −K6
12 0 0 K6 K1 −K7 −K2
13 K6 K6 0 −K7 K4 K7
21 0 0 −K6 −K2 K7 K1
(c)
11 22 23 12 13 21
11 R1 R1 R6 R2 R5 −R2
22 −R1 −R1 −R6 −R2 −R5 R2
33 0 0 0 0 0 023 R4 −R4 0 R3 0 R3
31 −R3 R3 0 R4 0 R4
12 R2 R2 −R5 −R1 R6 R1
Jaric (1987) derived the number of the elasic constant and the explicit forms of elastic freeenergy for icosahedral and pentagonal QCs. Socolar (1989) gave the elastic constant tensorsof the planar octagonal and dodecagonal QCs. Huet al (1993b) obtained those for 3D 8-,10- and 12-fold symmetries. Jaric and Nelson (1988) proposed an elastic theory of diffusescattering from QCs, within the framework of which the phonon and phason elastic constantscan be evaluated (Widom 1991, Leiet al 1998). The diffuse scattering located close to Braggreflections has been recently measured on a single grain of the AlPdMn icosahedral phase andthe ratio of two phason elastic constants is found to be equal to 0.5 (de Boisseuet al 1995,Boudardet al 1996). In addition to using the diffuse scattering from QCs, elastic constants ofsome icosahedral QCs have also been measured by the ultrasonic pulse–echo method (Reynolds
Physical properties of quasicrystals 17
Table 4. (Continued)(d)
11 22 23 12 13 21
11 K1 K2 0 K5 0 K5
22 K2 K1 0 −K5 0 −K5
23 0 0 K4 0 0 012 K5 −K5 0 6 0 K3
13 0 0 0 0 K4 021 K5 −K5 K3 0 0 6
(e)
11 22 23 12 13 21
11 R1 R1 0 R2 0 −R2
22 −R1 −R1 0 −R2 0 R2
33 0 0 0 0 0 023 0 0 0 0 0 031 0 0 0 0 0 012 R2 R2 0 −R1 0 R1
Table 5. Piezoelectric constants for 2D QCs with noncrystalline symmetries. In this table theindicesjk in the phason strainWjk are arranged in the order of 11, 22, 23, 12, 13 and 21.
5, 10, 8, 12 5m, 10mm, 8mm, 12mm
d(1) = 0 0 0 d
(1)14 d
(1)15 0
0 0 0 d(1)15 −d(1)14 0
d(1)31 d
(1)31 d
(1)33 0 0 0
4
d(1) = 0 0 0 0 d
(1)15 0
0 0 0 d(1)15 0 0
d(1)31 d
(1)31 d
(1)33 0 0 0
3
52, 1022, 822, 1222 5, 10
d(1) = 0 0 0 d
(1)14 0 0
0 0 0 0 −d(1)14 00 0 0 0 0 0
1
d(2) = d(2)111 −d(2)111 0 d
(2)112 0 d
(2)112
d(2)112 −d(2)112 0 −d(2)111 0 −d(2)1110 0 0 0 0 0
2
52,102m (with 2 ‖ x1) 5m,10m2 (withm ⊥ x1)
d(2) = d(2)111 −d(2)111 0 0 0 0
0 0 0 −d(2)111 0 −d(2)1110 0 0 0 0 0
1
d(2) = 0 0 0 d
(2)112 0 d
(2)112
d(2)112 −d(2)112 0 0 0 00 0 0 0 0 0
1
8, 12 82m,122m (with 2 ‖ x1)
d(2) = 0 0 d
(2)123 0 d
(2)113 0
0 0 −d(2)113 0 d(2)123 0
d(2)311 −d(2)311 0 d
(2)312 0 d
(2)312
4
d(2) = 0 0 d
(2)123 0 0 0
0 0 0 0 d(2)123 0
0 0 0 d(2)312 0 d
(2)312
2
8m2,12m2 (withm ⊥ x1) in other cases
d(2) = 0 0 0 0 d
(2)113 0
0 0 −d(2)113 0 0 0
d(2)311 −d(2)311 0 0 0 0
2
either d(1) = 0or d(2) = 0or d(1) = d(2) = 0
et al 1990), by Brillouin scattering (Vanderwalet al 1992), by the conventional tensile test(Yokoyamaet al 1995) and by the rectangular parallelepiped resonance method (Tanakaet al1996). However, the results seem to be inconsistent. It is desirable to accumulate moreexperimental data in order to understand the elastic properties of the material better.
In section 3.2 we calculated the number of independent components in the elastic constanttensors; we now determine the form of the elastic constant tensors. Clearly, the determinationof explicit forms for these independent components is much more complicated than counting
18 C Z Hu et al
Table 6. Piezoelectric constants for 3D QCs. In this table the indicesjk in the phason strainWjk
are arranged in the order 11, 22, 33, 23, 31, 12, 32, 13 and 21.
Piezoelectric constants
Point groups d(1) d(2)
23
0 0 0 d(1)14 0 0
0 0 0 0 d(1)14 0
0 0 0 0 0 d(1)14
1
0 0 0 d(2)123 0 0 d
(2)132 0 0
0 0 0 0 d(2)123 0 0 d
(2)132 0
0 0 0 0 0 d(2)123 0 0 d
(2)132
2
43m
0 0 0 d(1)14 0 0
0 0 0 0 d(1)14 0
0 0 0 0 0 d(1)14
1
0 0 0 d(2)123 0 0 d
(2)123 0 0
0 0 0 0 d(2)123 0 0 d
(2)123 0
0 0 0 0 0 d(2)123 0 0 d
(2)123
1
432 d(1) = 0
0 0 0 d(2)123 0 0 −d(2)123 0 0
0 0 0 0 d(2)123 0 0 −d(2)123 0
0 0 0 0 0 d(2)123 0 0 −d(2)123
1
m3, m3m235, m35
d(1) = d(2) = 0
their number. From equation (3.9) we can see that the transformation properties ofCijkl ,KijklandRijkl follow directly from those ofEij (Tij ) andWij (Hij ). If we find the precise componentsof Eij (Tij ) andWij (Hij ) that transform under the same constituent representation, we canconstruct all the invariants formed by their combinations and then establish the independentcomponents ofCijkl ,Kijkl andRijkl . As an example, we still consider a QC withC8v symmetry.In this caseα = 2π
8 , 2α = 4π8 ,β = 6π
8 ,β−α = 4π8 andβ+α = π (cf equation (3.15)). Here, it
should be pointed out that there are two generators of the symmetry group C8v, but, nevertheless,the reduced form given in equation (3.15) holds equally true for this case. Consequently, wefind that two 1D blocks and two 2D blocks make up the invariant subspaces of the elastictensor representation associated with the phonon field. In group-theoretical jargon, this tensorrepresentation can be decomposed into irreducible representations:
{(05 + 01)× (05 + 01)}S = 201 + 05 + 06 (3.20)
where{. . .}S denotes the symmetric product representation. This means thatE11 + E22 andE33 form two 1D subspaces (the identity representation01). They are two linear invariants.E11−E22 amd 2E12 form one 2D subspace (06) giving one quadratic invariant (scalar product)
(E11− E22) · (E11− E22) + (2E33) · (2E33) = (E11− E22)2 + 4E2
12. (3.21)
E13 andE23 form another 2D subspace (05) giving another quadratic invariant
E213 +E2
23. (3.22)
Thus, there are a total of five quadratic invariants due to the phonon field
(E11 +E22)2, E2
33, (E11 +E22)E33, E213 +E2
23, E11E22− E212.
(3.23)
Of these, the first three are products of two linear invariants:E11 + E22 andE33. Fromequation (3.23) it follows that non-vanishing components are
C1111= C2222= C11, C1133= C2233= C3311= C3322= C13, C3333= C33,
C2323= C2332= C3223= C3232= C1313= C1331= C3113= C3131= C44,
C1122= C2211= C12, C1212= C1221= C2112= C2121= C66 = 12(C11− C12)
(3.24)
Physical properties of quasicrystals 19
which can be written in the matrix form
C =
C11 C12 C13 0 0 0C12 C11 C13 0 0 0C13 C13 C33 0 0 00 0 0 C44 0 00 0 0 0 C44 00 0 0 0 0 C66
5
(3.25)
where the subscript 5 stands for the number of independent components. The correspondencesbetween the index pairs and single indices are, as usual,
(ij) = 11 22 33 23 31 12
i = 1 2 3 4 5 6.(3.26)
Similarly, for the phason field six components ofWij transform under
(05 + 01)× 07 = 03 + 04 + 06 + 07. (3.27)
From equation (3.15) we conclude thatW11 − W22 andW21 + W12 form two different 1Dsubspaces (03 and04) giving two quadratic invariants
(W11−W22)2, (W21 +W12)
2. (3.28)
Two pairs (W11 +W22,W21−W12) and (W13,W23) form two different 2D subspaces (06 and07) giving two quadratic invariants
(W11 +W22)2 + (W21−W12)
2, W 213 +W 2
23. (3.29)
Thus, there are a total of four quadratic invariants due to the phason field. They are listed inequations (3.28) and (3.29). The non-vanishing components are
K1111= K2222= K1, K1122= K2211= K2, K1221= K2112= K3,
K2121= K1212= K1 +K2 +K3, K1313= K2323= K4.(3.30)
The corresponding matrix form is
K =
K1 K2 0 0 0 0K2 K1 0 0 0 00 0 K4 0 0 00 0 0 K1 +K2 +K3 0 K3
0 0 0 0 K4 00 0 0 K3 0 K1 +K2 +K3
4
(3.31)
where the double indices labelling the phason strains are arranged in the order of 11, 22, 23, 12,13, 21. Finally, notice that the pairs (E11−E22, 2E12) and (W11 +W22,W21−W12) transformaccording to the same representation (06). This means that there exists an invariant
(E11− E22)(W11 +W22) + 2E12(W21−W12) (3.32)
couplingEij toWij . Non-vanishing components are
R1111= R1122= −R2211= −R2222= R1221= R2121= −R1212= −R2112= R (3.33)
with the matrix form
R =
R R 0 0 0 0−R −R 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 −R 0 R
1
. (3.34)
20 C Z Hu et al
Therefore, it can be seen that there are ten quadratic invariants and hence ten independent elasticconstants for 8mm symmetry. Among them, five components (nC = 5) are due to the phononfield, four constants (nK = 4) due to the phason field, and one constant (nR = 1) associatedwith the phonon–phason coupling. The results coincide with those given in equation (3.19). Inthe same way, we can determine the explicit form of the elastic constants for QCs with varioussymmetries. All the results are listed in table 4.
3.4. Thermal expansion and piezoelectricity tensors
Besides the elastic tensor, there are two types of physical property tensors (thermal expansionand piezoelectric tensors) for which one must consider the contributions from both the phononand phason fields. The form of expansion tensorsα
(1)ij andα(2)ij can be deduced from general
consideration of symmetry. For QCs with non-crystallographic symmetries a vector inVE anda vector inVI transform under different representations of the point group (Lubensky 1988).For example, in the case of 2D QCs they transform according to02 +01 and02
′, respectively.Here02 and02
′ are two non-equivalent 2D representations and01 a 1D representation. Inthe case of icosahedral QCs they transform according to03 and03
′ where03 is a 3D vectorrepresentation and03
′ a 3D non-vector-like representation. Consequently, the componentsα(1)ij transform under
(02 + 01)× (02 + 01) = 02 × 02 + 202 + 01 (3.35)
for 2D QCs and
03× 03 (3.36)
for icosahedral QCs. The componentsα(2)ij transform under
(02 + 01)× 02′ = 02 × 02
′ + 02′ (3.37)
for 2D QCs and
03× 03′ (3.38)
for icosahedral QCs. From the character orthogonality theorem (Lax 1974), we can infer thatthe identity representation appears once in0A × 0A, but does not appear at all in0A × 0Bwhere0A and0B are two non-equivalent irreducible representations. It follows immediatelythat there are no non-vanishing components in the tensorα
(2)ij , i.e.
α(2)ij = 0 (3.39)
but two independent components inα(1)ij for 2D QCs and one independent component inα(1)ij
for icosahedral QCs. Written out, the surviving scheme of constants is
α(1) =α(1)11 0 0
0 α(1)11 0
0 0 α(1)33
2
(3.40)
for 2D QCs and
α(1) =α(1)11 0 0
0 α(1)11 0
0 0 α(1)11
1
(3.41)
for icosahedral QCs.To determine the form of the piezoelectric tensors, we still consider the example of
octagonal QCs with 8mm symmetry. From equations (3.15) and (3.16) we can see that the
Physical properties of quasicrystals 21
transformation properties ofd(1)ijk , d(2)ijk (d(1)ijk, d
(2)ijk) follow directly from those ofDi (Fi), Tjk
(Ejk) andHjk (Wjk). As we know, the electric displacement vectorDi transforms under thevector representation (05+01) while the phason strainsWij transform under03+04+06+07 (cfequation (3.27)). Since there is no identity representation in(05+01)×(03+04+06+07), thecomponentsd(2)ijk = 0. For the componentsd(1)ijk , using a method similar to that in section 3.3,we find thatE11 + E22, E33 andD3 transform under the same representation (01) giving twoinvariants
D3(E11 +E22), D3E33. (3.42)
Thus, we obtain the non-vanishing components
d(1)311= d(1)322= d(1)31 , d
(1)333= d(1)33 . (3.43)
Two pairs (D1,D2) and (E13,E23) transform under the same representation (05) giving anotherinvariant
D1E13 +D2E23. (3.44)
Then the corresponding non-vanishing component is
d(1)113= d(1)223= d(1)15 . (3.45)
Hence the piezoelectric tensord(1) has the form
d(1) = 0 0 0 0 d
(1)15 0
0 0 0 d(1)15 0 0
d(1)31 d
(1)31 d
(1)33 0 0 0
3
(3.46)
with three independent constants, which coincide with those given in section 3.3. In this mannerwe can determine the form of the piezoelectric tensors for all QCs. All the results are given intables 5 and 6. The knowledge of the physical properties beyond elasticity is very incompletealthough some preliminary investigations have been made in this field (Brandmuller and Claus1988, Fujiwaraet al 1994, Yanget al 1996, Huet al 1997a). Most of the physical propertiespredicted in QCs still remain to be confirmed experimentally.
4. Generalized elasticity theory
4.1. Phonon and phason fields
There are two descriptions used for studying the structural and elastic properties of QCs (Levineet al 1985, Lubenskyet al 1985, 1986, 1988, Bak 1985a, Kaluginet al 1985b, Troian andMermin 1985, Jaric 1985, Duneau and Katz 1985, Socolaret al 1986, Gahler and Rhyner1986). One is the density-wave description and the other is the unit-cell description. In thedensity-wave description based on Landau theory the density of the ordered, low-temperaturephase can be written
ρ(r) =∑GεLR
ρG exp(iG · r) (4.1)
whereG is a reciprocal vector andLR is the reciprocal lattice. EachρG is a complex numberwith an amplitude|ρG| and a phase8G. Sinceρ(r) is real,|ρG| = |ρ−G| and8G = −8G.Consequently, the free energyF can be expanded in a power series inρ. For example, thekthpower ofρ yields the following term inF :
F (k) = Ak∑GnεLR
1
[∑n
Gn
]cos
[∑n
8Gn
]5|ρGn
|. (4.2)
22 C Z Hu et al
The factor1(x) = δx,0 ensures that only terms where6Gi = 0 contribute to the sum. Theequilibrium ordered state is given by the values of|ρG| and8G that minimizeF . In practice,it is sufficient to consider a small finite subset{ρG} of the Fourier components. Obviously,the subset{ρG} must include Fourier components associated with theNR reciprocal vectorsthat form a basis of the reciprocal latticeLR. In addition,{ρG} must include the inversecomponents{ρ−G} plus any vectors that can be obtained from the minimal set by point-symmetry operations. IndD QCs,NR = nid whereni is the number of incommensuratelengths associated with each lattice-vector direction. For the QCs observed to dateni = 2.For periodic crystalsni = 1. In addition, Huet al (1994) had theoretically investigated thecase ofni = 3. From equation (4.2) it is clear that for any set ofGn that satisfies6Gn = 0,minimization ofF with respect to|ρG|and8G leads to a minimum-energy state with constraintson the8G (68G = γ = const.). For planar pentagonal QCs the five vectors
Gn = G[cos(2πn/5), sin(2πn/5)] n = 0, 1, . . .4 (4.3)
determine the fivefold symmetry and generate the reciprocal lattice. Obviously, the ratio ofG0 andG1 +G4 is not rational (ni = 2). These vectors are not, however, independent because6Gn = 0. Moreover, in the minimum-energy state the phases associated withGn satisfy68n = γ = const. This means that only four of theGn and8n are independent (NR = 4).We can parametrize the four independent components of8n by two two-component fields,uandw, according to
8Gn= Gn · u + aG⊥n ·w + γ /5 n = 0, 1, . . .4 (4.4)
whereG⊥n is the complementary vector and will be calculated with the aid of group theory.For QCs with C5 symmetry four independent vectorsGn span a reducible representation0 ofC5. The reduction is
0 = 02 + 0′2 (4.5)
where02 and0′2 are 2D representations (Lubensky 1988, Steurer 1990, Huet al 1993a, b).One,02, is the vector representation. The other,0′2, is not. The vectorsGn transform under02, and the complementary vectorsGn
⊥ transform under0′2. Since0′2 is generated by acounterclockwise(6π)/5 rotation,G⊥n = G〈3n〉 where〈3n〉means 3nmod 5. It is convenientto treatG‖n = Gn andG⊥n as components of a higher-dimensional vector:
Gn = G‖n + αG⊥n (4.6)
where + denotes the direct sum. Then equation (4.4) can be rewritten as
8Gn= G · u + γ /5 (4.7)
with u = u‖ + u⊥ = u +w.It can be seen that QCs have two types of elastic hydrodynamic variables: one is the
phonon variableu and the other is the phason variablew. Phonons and phasons are low-energy hydrodynamic modes associated with quasiperiodic broken translational symmetry.At long wavelengths phonons correspond to uniform translations, and phasons correspond torearrangements of atoms from one potential valley to another. Such rearrangements induced byw can be better understood in the unit-cell description of QCs (Socolaret al1986). Consider so-called ‘worms’ in the tiling. A ‘worm’ is a linear chain composed of thin and thick rhombuses.In general, a uniform shift inw causes a number of criss-crossing worms to flip in such a waythat the matching rules are maintained throughout. Spatial variations inw result in segmentsof worms being flipped, causing matching rule violations. Uniformu andw do not change thefree energy; spatially nonuniformu andw do, however. If an analytic expansion of elastic freeenergy is possible, its first term at long wavelength will be quadratic in the spatial gradients
Physical properties of quasicrystals 23
of u andw. Once the elastic constant tensors have been derived for a given QC, its elasticenergyF can be easily determined (see section 3.3).
Here two points are worth noting. First, a real quasicrystalline material is a 3D solid.Therefore,u is a three-component vector andw is a two-component vector for 2D QCs.u isstill a three-component vector, butw is a one-component vector for 1D QCs.u andw are boththree-component vector for icosahedral QCs. Second, QCs are unique among systems withcontinuous broken symmetries. They can exhibit two generically different types of energeticsand dynamics associated with strains of their elastic variables (Socolaret al1986, Frenkelet al1986, Lubensky 1988, Steinhardt and DiVincenzo 1991). In the density-wave pictureu andw are regarded as continuum hydrodynamic variables. By the standard theory of elasticity theelastic energy is proportional to the square of their gradients. In the unit-cell picture spatialvariations inw give rise to matching rule violations. The elastic energy associated with a givenconfiguration with nonuniformw is thenF = Nmε whereNm is the number of matching ruleviolations andε is an energy associated with each matching rule violation. Since the numberof matching rule violations is proportional to|δw|, F ∼ |δw|. Thus, one might imagine atransition between these two types of dynamics. Such transitions have been discussed by Jeongand Steinhardt (1993) who applied the phason dynamics of a tiling model to demonstration ofthe transition at a temperatureTc from a locked phase (in whichF ∼ |δw|) whenT < Tc toan unlocked phase (in whichF ∼ (δw)2 ) whenT > Tc. The unit cells provide a template forlocal atomic configurations, but do not determine the dynamics. Here and hereafter we adoptthe viewpoint of the density-wave description.
4.2. General expressions for elastic equations
In the higher-dimensional description of QCs, andD QS with Fourier modulus of rankn canbe generated by intersecting annD spaceV (V = VE + VI ) by the 3D physical subspaceVE .Consequently, annD displacement vectoru in V , when projected uponVE andVI , becomesa direct sum:
u = u‖ + u⊥ = u +w (4.8)
whereu is an 3D vector inVE (phonon variable) andw is an (n − 3)D vector inVI (phasonvaviable). It should be noted that bothu andw are functions of the position vector in thephysical spaceVE (De and Pelcovits 1987). Then we have
∇u = ∇u +∇w (4.9)
where∇ = ei∇i with ∇i = ∂∂xi= ∂i , is a differential operator relative to the position vector.
∇u can be decomposed into symmetric and antisymmetric parts. The antisymmetric partdescribes a rigid rotation that does not change the elastic energy. Since there is an energy costassociated with the antisymmetric derivative ofw, the elastic energy depends on all componentsof the gradient ofw (Lubensky 1988). Thus, the elastic energy depends only on the followingstrains:
Eij = 12(∂jui + ∂iuj )
Wij = ∂jWi.(4.10)
It is easy to verify the compatibility equation (Dinget al 1994):
−eijkelmn∂j ∂mEkn = 0
−eijkelmn∂j ∂mwkn = 0(4.11)
24 C Z Hu et al
whereeijk is the alternator symbol. Furthermore, the elastic energy density can be expandedin terms of the Taylor series in the vicinity ofEij = 0 andWij = 0 to second order:
F(Emn,Wmn) = 1
2
[∂2F
∂Eij ∂Eki
]0
EijEkl +1
2
[∂2F
∂Wij ∂Wkl
]0
WijWkl
+1
2
[∂2
∂Eij ∂Wkl
]0
EijWkl +1
2
[∂2
∂Wij ∂Ekl
]0
WijEkl
= 12CijklEijEkl + 1
2KijklWijWkl + 12RijklEijWkl + 1
2R′ijklWijEkl (4.12)
where
Cijkl =[
∂2F
∂Eij ∂Ekl
]0
(4.13)
are quadratic elastic constants in the classical elasticity theory withCijkl = Cklij = Cjikl =Cijlk. We can denote allCijkl by a symmetric matrix [C]. Similarly,
Kijkl =[
∂2F
∂Wij ∂Wkl
]0
(4.14)
are elastic constants of the phason field withKijkl = Kklij , which can be also decoded by asymmetric matrix [K].
Rijkl =[
∂2F
∂Eij ∂Wkl
]0
R′ijkl =[
∂2F
∂Wij ∂Ekl
]0
(4.15)
are the elastic constants associated with the phonon–phason coupling. Obviously,Rijkl =Rjikl, R
′ijkl = R′ij lk, R
′klij = Rijkl , but Rijkl 6= Rklij , R
′ijkl 6= R′klij . We denote them by
matrices [R] and [R′] with [R′] = [R]T . These four matrices compose a matrix [C,K,R]:
[C,K,R] =[
[C] [R][R′] [K]
]=[
[C] [R][R]T [K]
](4.16)
(Ding et al 1994, Yanget al 1993). With these notations, we can write the elastic energy in acompact form:
F = 1
2[E,W ]
[[C] [R][R]T [K]
] [E
W
]. (4.17)
The movement of atoms through the barrier needs some forces when the unit cellsrearrange. Thus, besides the conventional body force densityf and surface force densityt in VE , a generalized body force densityg and surface force densityh in VI should beintroduced in the elasticity of QCs. Similarly, in addition to the conventional stress tensorT ,there is another stress tensorH with componentsHij along thexi direction inVI acting onthe surface orthogonal to thexj direction inVE . If n is the outward unit normal vector to thesurface, then we have
ti = Tijnj hi = Hijnj . (4.18)
As Lubenskyet al (1985) pointed out, in the case of QCs the momentum theorem possessesthe form
d
dt
∫V
ρ(u + w) dV =∫V
(f + g) dV +∫s
(t + h) ds (4.19)
whereV is an arbitrary volume inVE andS is the boundary surface ofV . Application of theGauss theorem to equation (4.19) leads to two equations of motion
∂jTij + fi = ρui ∂jHij + gi = ρwi (4.20)
Physical properties of quasicrystals 25
and two static equilibrium equations
∂jTij + fi = 0 ∂jHij + gi = 0. (4.21)
The angular momentum theorem associated with the phonon field has the formd
dt
∫V
r‖ × ρudV =∫V
r‖ × f dV +∫S
r‖ × t dS. (4.22)
Hence, by the Gauss theorem and equation (4.20) we obtain
Tij = Tji . (4.23)
This means that the phonon stress tensor is symmetric. Sincer‖ andw(g,h) transform underdifferent representations of the point group, more precisely, the former transforms like a vector,but the latter does not, the product representations,r‖ × w, r‖ × g, r‖ × h do not contain anyvector representations. This implies that for the phason field there is no equation analogous to(4.22), from which it follows that, generally,
Hij 6= Hji. (4.24)
By an argument such as that given in classical elasticity theory we obtain
Tmn = ∂F
∂EmnHmn = ∂F
∂Wmn
. (4.25)
Substituting equation (4.12) into the above equations gives the generalized Hooke’s law
Tij = CijklEkl +RijklWkl
Hij = RklijEkl +KijklWkl
(4.26)
which can be expressed in the matrix form[T
H
]=[
[C] [R][R]T [K]
] [E
W
]. (4.27)
Substituting (4.26) into (4.21) gives the non-homogeneous partial differential equationssatisfied byu andw:
Cijkl∂j ∂luk +Rijkl∂j ∂lwk + fi = 0
Rklij ∂j ∂luk +Kijkl∂j ∂lwk + gi = 0.(4.28)
To apply the generalized elasticity theory to the case of icosahedral QCs, it should benoted that the elastic constants depend on the choice of the coordinate system. Unfortunately,a particular choice for the coordinate system is somewhat arbitrary, and at least two choicesappear in the literature. A coordinate system can be chosen with thez-axis pointing towards avertex of an icosahedron (Levineet al1984, Bak 1985b, Dinget al1992). Another coordinatesystem has thez-axis normal to an edge of an icosahedron (Lubenskyet al1985, Jaric 1987). Inthe framework of the former coordinate system it can be seen thatCijkl has the same expressionas for any isotropic solid:
Cijkl = λδij δkl +µ(δikδjl + δilδjk) (4.29)
and [K] and [R] are
[K]=
K1 0 0 0 K2 0 0 K2 00 K1 0 0 −K2 0 0 K2 00 0 K1 +K2 0 0 0 0 0 00 0 0 K1−K2 0 K2 0 0 −K2
K2 −K2 0 0 K1−K2 0 0 0 00 0 0 K2 0 K1 −K2 0 00 0 0 0 0 −K2 K1−K2 0 −K2
K2 K2 0 0 0 0 0 K1−K2 00 0 0 −K2 0 0 −K2 0 K1
26 C Z Hu et al
(4.30)
[R] = R
1 1 1 0 0 0 0 1 0−1 −1 1 0 0 0 0 −1 00 0 −2 0 0 0 0 0 00 0 0 0 0 −1 1 0 −11 −1 0 0 1 0 0 0 00 0 0 −1 0 −1 0 0 10 0 0 0 0 −1 1 0 −11 −1 0 0 1 0 0 0 00 0 0 −1 0 −1 0 0 1
. (4.31)
Substituting equations (4.29) and (4.30) into equations (4.26) and (4.28), Dinget al (1993)gave the practical elastic equations of icosahedral QCs.
5. Elasticity theory of dislocation
5.1. Dislocations in QCs
Soon after their discovery, QCs were found to contain structural defects. Studies of the defectsin QCs have attracted extensive attention because of their importance not only to structuralanalyses but also to a deep understanding of the mechanical and physical properties of QCs.Dislocations were observed in Al–Mn-Si, Al–Li–Cu and Al–Cu–Fe icosahedral phases byelectron microscopy (Hiraga and Hirabayashi 1987, Chenet al 1987, Devaud-Rzepskiet al1989, Ebalard and Spaepen 1989, Yuet al 1992). In addition, twins (Koskenmakiet al 1986,Dai and Urban 1993, Shield and Kramer 1994), stacking faults (Menonet al 1989, Daiet al1991), discommensurations (Jianget al 1991) and domains (Audieret al 1990, Tsudaet al1993) have been found experimentally. As is the case for crystals, defects in QCs will affecttheir growth, phase transitions, mechanical behaviour, etc. Therefore, extensive work has beenstimulated in this field (Levineet al 1985, Lubenskyet al 1985, Socolaret al 1989, Kleman1992, Yuet al 1997). As a result, such studies have opened a new research field in defectphysics. Dislocations are important defects in QCs, to which our discussion is restricted. AQS can be described as a section of a higher-dimensional periodic structure. By use of thisfact, we can extend the concept of the Burgers vector in ordinary crystals to QCs. A Burgersvectorb corresponds to displacements inu andw that leave the8Gn
invariant modulo 2π . Todetermine the Burgers vectors we consider the example of pentagonal QCs. First, we constructthe vectors
R‖n = R[− sin
2πn
5, cos
2πn
5
]n = 0, 1 . . .4 (5.1)
orthogonal to the vectorsGn in equation (4.3). Next, we construct 4D vectors
Rn = R‖n + βR⊥n (5.2)
whereβ is an arbitrary scale factor andR⊥n = R‖〈3n〉 are orthogonal to the complementary
vectorsG⊥n = G‖〈3n. Then we have
8n,m = Gn · Rm = GR{sin[2π(n−m)/5] + αβ sin[6π(n−m)/5]}. (5.3)
Requiring that8n,(n+2) = −8n,(n+3) = 2π and8n,n = 8n,(n+1) = 8n,(n+4) = 0 (Levineet al1985) yields
GR = 8π
5sin
4π
5αβ = sin(2π/5)
sin(4π/5)= τ = 1 +
√5
2. (5.4)
Physical properties of quasicrystals 27
Setβ = 1 andα = τ . Then the extended basis of the Burgers-vector lattice is obtained byforming the direct sum of vectorsR‖n andR⊥n . Alternatively, we consider a closed loopCwhich encircles the dislocation core. The integral of the gradient of the phases aroundC isnot zero, but an integral multiple of 2π , i.e. the value of this integral is∮
c
∇8n(r) · dl = Gn ·∮c
∇u(r) · dl = m2π (5.5)
which carries a Burgers vector
b =∮c
∇u · dl. (5.6)
Here it should be noted that:
(1) Any Burgers vectorb must have a parallel componentb‖ in VE and a perpendicularcomponentb⊥ in VI (b = b‖ + b⊥). There is nob with b‖ 6= 0 andb⊥ = 0, or b⊥ 6= 0andb‖ = 0. The physical implication of this fact is that dislocations in QCs, unlike thosein periodic crystals, cannot be interpreted simply in terms of insertions or removals ofhalf-planes of atoms.
(2) Each dislocation in a QC is accompanied by a phason strain in addition to the usual phononstrain.
(3) The elastic fields induced by dislocations in QCs can be calculated using a method similarto that in periodic crystals. Explicit calculations have been carried out by De and Pelcovitz(1987) for pentagonal QCs in two dimensions and by Dinget al (1995b) for QCs in threedimensions, as discussed in section 5.2.
A Volterra process for the construction of a dislocation in a crystal consists of (1) cuttingthe crystal along the arbitrary surface bounded by the location, (2) rigidly translating the twolips by a Burgers vector, (3) filling and removing matter and (4) gluing the lips together (Steeds1973). As we know, periodic crystals have translational symmetry and the Burgers vector ofa perfect dislocation is a lattice vector. The Volterra process restores the lattice everywhereoutside the dislocation core except for continuous deformations. In QCs a Volterra processcannot be performed in the same way because of the absence of translational symmetry. Indeed,removing a piece of a trail and gluing the lips together leads to vertices outside the originallocal isomorphism (LI) class (Bohsug and Trebin 1989). To solve this problem we have toperform the Volterra process in annD spaceV from which the QS is generated. A dislocationline L in V is, by definition, a singularity such that when circumnavigating about it along anyloop, the sum of the elastic displacements is a constantb (the Burgers vector):
b =∮
du =∮∇u · dl (b = b‖ + b⊥). (5.7)
The physical dislocation inVE is the intersectionL‖ of L with VE , i.e.
L = L‖ + VI (5.8)
whereL is a (n−2)D superline andL‖ is an 1D line in the physical spaceVE . Now we can taketwo steps to introduce the Volterra process for the construction of a dislocation in QCs. First,cut the hyperlattice inV along a (n − 1) hypersurface
∑forming two lips
∑1,∑
2, rigidlytranslate the lip
∑1 relative to the lip
∑2 byb‖, remove extra matter and glue the lips together.
Let (L, b‖) denote the configuration resulting from this step. The intersection ofL with VEis the physical dislocationL‖ , and the intersection of
∑with VE is a stacking fault. Second,
we shift the lips by a vectorb⊥. Sinceb = b‖ + b⊥ is a lattice vector in higher-dimensionalspace, the lips of a cut match again after these two consecutive shifts. Thus, in the final state
28 C Z Hu et al
the stacking fault disappears, but mistakes appear. From the above description it is clear thatthe first step results in the phonon strain field and the second step results in the phason strainfield around the dislocation (Dinget al 1998). So far we have completed the Volterra processfor introducing a dislocation in QCs.
5.2. Elastic fields induced by general dislocations
In order to find the elastic fields induced by dislocations in QCs, we must solve theinhomogeneous partial differential equations (4.28) with given boundary conditions as oftenused in the elasticity theory of dislocations in crystals. In the derivation that follows we assumea QC bodyτ to be an infinite, homogeneous and free body. Thus, the boundary condition
Tijnj = Hijnj = 0 (5.9)
holds whenx → ∞. According to the eigenstrain method (Mura 1987), a subdomainτ ′ inthe bodyτ is subjected to eigenstrainsE∗ij andW ∗ij during the Volterra process. The actualstrains in the domainsτ , Eij andWij , are the sum of the eigenstrainsE∗ij andW ∗ij and elasticstrainsE′ij andW ′ij , i.e.
Eij = E′ij +E∗ij Wij = W ′ij +W ∗ij . (5.10)
Since the elastic strain fieldsE′ij andW ′ij induced by the dislocation are related to the internalstress fieldTij andHij by the generalized Hook’s law (4.26), we obtain
Tij = CijklEkl +RijklWkl − CijklE∗kl − RijklW ∗klHij = RklijEkl +KijklWkl − RklijE∗kl −KijklW ∗kl .
(5.11)
Substituting equation (5.11) into (4.21) and consideringfi = gi = 0, we have
Cijkl∂j ∂luk +Rijkl∂j ∂lwk = Cijkl∂jE∗kl +Rijkl∂jW∗kl
Rklij ∂j ∂luk +Kijkl∂j ∂lwk = Rklij ∂jE∗kl +Kijkl∂jW∗kl .
(5.12)
By comparing equation (5.12) with (4.28) one can see that the contribution ofE∗kl andW ∗kl tothe equilibrium equations is similar to that of the two-body forcesXi andYi :
Xi = −Cijkl∂jE∗kl − Rijkl∂jW ∗klYi = −Rklij ∂jE∗kl −Kijkl∂jW ∗kl .
(5.13)
Upon Fourier transformation equation (5.12) becomes
Mikuk +Rikwk = XiRTikuk +Nikwk = Y i (5.14)
where an overbar represents the Fourier transform of a function, i.e.
f (x) =∫ ∞−∞
f (k) exp(ik · x) dk
f (k) = 1
8π3
∫ ∞−∞
f (x) exp(−ik · x) dx(5.15)
where
Mik = Cijklkj kl Rik = Rijklkj klRTik = Rklij kj kl Nik = Kijklkj kl (5.16)
and
Xi = −iCijklkjE∗kl − iRijklkjW
∗kl
Y i = −iRklij kjE∗kl − iKijklkjW
∗kl .
(5.17)
Physical properties of quasicrystals 29
Equation (5.14) can be written in a compact form:
Aαβ(k)Vβ(k) = Zα(k) (5.18)
if we defineAαβ, Vα
andZα
as
Aαβ = δαi (δβj Mij + δβ−3j Rij ) + δα−3
i (δβ
j Rji + δβ−3j Nij )
Vα(k) = δαi ui(k) + δα−3
i wi(k)
and
Zα(k) = δαi Xi(k) + δα−3
i Y i(k) (5.19)
where
δαi ={
1 α = i0 α 6= i δα−3
i ={
1 α − 3= i0 α − 3 6= i. (5.20)
Equation (5.18) is a set of inhomogeneous algebraic equations, the solution of which can beexpressed in terms of Green functions as
Vα(k) = Gαβ
(k)Zβ(k) (5.21)
where
Gαβ(k) = [Aαβ(k)]−1 (5.22)
and
Zβ(k) = 1
8π3
∫ ∞−∞
Zβ(x′) exp(−ik · x′) dx′. (5.23)
When we transform equation (5.21) from Fourier space back to coordinate space, we find thegeneral expressions of the elastic displacement fields induced by dislocations in QCs (Dinget al 1994, 1995b, Yaoet al 1997):
V α(x) =∫ ∞−∞
Gαβ(x− x′)Zβ(x′) dx′ (5.24)
with Green functions
Gαβ(x− x′) = 1
8π3
∫ ∞−∞
Gαβ(k) exp[ik · (x− x′)] dk. (5.25)
Substituting equations (5.13) and (5.19) into (5.24) and integrating by parts, we have
V α(x) = −∫ ∞−∞
∂jGαβ(x− x′)(δβi Cijkl + δβ−3
i Rklij )E∗kl(x
′) dx′
−∫ ∞−∞
∂jGαβ(x− x′)(δβi Rijkl + δβ−3
i Kijkl)W∗kl(x
′) dx′. (5.26)
The following is noteworthy. (1) The body forcesXi andYi (equation (5.13)) both act atthe pointx ′ in PE , but the directions ofXi andYi are along the axes inPE andPI , respectively.(2) Gαβ(x− x′) represents theα component of the displacements produced by a unit-pointbody force along theβ direction acting at the pointx ′i in PE . (3) The elastic constants inthe dislocation coordinate system are not, in general, the same as those in the conventionalcoordinate system of QCs, and a coordinate transformation must be performed accordingly.
30 C Z Hu et al
5.3. Elastic fields induced by straight dislocations
As we know, besides the Green function method given in section 5.2, there are other methodsin the case of ordinary crystals, such as Eshelby’s method (Eshelbyet al 1953, Hirth andLothe 1968) and Stroh’s method (Stroh 1958, 1962, Baconet al 1978), which may have anadvantage over the Green method for some actual calculations. In the following, we generalizethese two methods to the case of QCs and give general expressions for the elastic fields ofstraight dislocation lines.
5.3.1. The generalization of Eshelby’s method (Ding et al 1995a).From the generalizedelasticity theory of QCs described in the preceding section, we know that the linear partialdifferential equations satisfied by the phonon and phason displacementsu(x) andw(x) areas follows (we assumefi andgi in equation (4.28) to be zero):
CiJkL∂J ∂Luk +RiJkL∂J ∂Lwk = 0
RkLiJ ∂J ∂Luk +KiJkL∂J ∂Lwk = 0(5.27)
where we choose a dislocation coordinate system in which thex3-axis is along the straightdislocation line in the physical space. It should be noted that the elastic constant tensorsCijkl, Kijkl andRijkl in equation (5.27) all refer to the dislocation coordinate system and thesubscripts with small letters(i, k) take the values 1, 2 or 3, while the capital letters(J, L) takethe values 1 or 2. Using
V β(x) = δβk uk(x) + δβ−3k wk(x) (5.28)
(see equation (5.19)) and setting
Bαβ
jl = δαi (δβk Cijkl + δβ−3k Rijkl) + δα−3
i (δβ
k Rklij + δβ−3k Kijkl) (5.29)
equation (5.27) can be written in a compact form:
Bαβ
JL∂J ∂LVβ(x) = 0. (5.30)
According to Eshelbyet al (1953), equation (5.30) has solutions of the type
V β(x) = Aβf (η) η = x1 + px2 (5.31)
wherep is an undetermined parameter which we find below.If we substitute equation (5.31) into (5.30) and introduce a matrix with the elements
aαβ = Bαβ11 + (Bαβ12 +Bαβ21 )p +B22p2 (5.32)
then we obtain a set of linear algebraic equations forA given by
aαβAβ = 0. (5.33)
It follows that the parameterp in equation (5.31) is determined by the condition
det|aαβ | = 0 (5.34)
which is an algebraic equation forp. Since the elastic energy is always positive and thecoefficients are all real, the roots of equation (5.34)p(n) are necessarily complex and occurin conjugate pairs. Following the analogous precedure proposed by Eshelbyet al (1953), thedisplacements induced by an infinite straight dislocation line parallel to the position of thex3-axis can be expressed in the form
V β(x) = Re
[ 6∑n=1
1
±2π iAβ(n)D(n)lnη(n)
](5.35)
Physical properties of quasicrystals 31
where Re means that only the real part is to be taken, the sign of 2π i is taken to be the same asthe sign of the imaginary part ofp(n). The constantsD(n) are determined by the followingequations:
Re
[ 6∑n=1
Aβ(n)D(n)
]= bβ
Re
[ 6∑n=1
(Bαβ
21 +Bαβ22p(n))Aβ(n)D(n)
]= 0
(5.36)
wherebβ are the components of the Burgers vectorb of the dislocation. Furthermore, if weintroduce the symbol
Mαj = δαi Tij + δα−3
i Hij (5.37)
the generalized Hook law (equation (4.26)) can be written as
Mαj = Bαβjl ∂lV β(r). (5.38)
Using the same method as Foreman (1955), the elastic energy per unit length of the dislocationcan be easily obtained as
W = kb2
4πln
(R
r0
)with
kb2 = bαIm
[∑n
±(Bαβ21 +Bαβ22p(n))Aβ(n)D(n)
]. (5.39)
5.3.2. The generalization of Stroh’s method (Ding et al 1995c).The anisotropic elasticitytheory of dislocation in crystals developed by Stroh (1958, 1962) has played an importantpart in the theory of defects in solids. Baconet al (1978) provided an overview on the Strohtheory in their review article in which the orthogonality, completeness and invariance relationsfor the Stroh eigenvectors, and the integral formalism were derived in considerable detail. Inwhat follows, we extend the six-dimensional (6D) formalism of Stroh to a higher-dimensionalone and then give an integral representation of the elastic fields induced by dislocations inQCs, which may be more suitable for numerical calculations. For definiteness we considericosahedronal QCs.
Since the elastic energy (equation(4.17)) is positive, i.e.
F = 1
2[Eij ,Wij ]
[Cijkl RijklRklij Kijkl
] [EklWkl
]> 0 (5.40)
the symmetric matrix
[C,K,R] =[Cijkl RijklRklij Kijkl
](5.41)
is always positive. The elements of [C,K,R] are none other thanBαβjl in equation (5.30).It follows that there exists a real non-zero vectora in VE and a real non-zero vectorC inV = VE + VI such that
CαajBαβ
jl alCβ > 0. (5.42)
This means that the matrix
ajBαβ
jl al = (aa)αβ = (aa) (5.43)
32 C Z Hu et al
is positive definite. This results guarantees the existence of the inverse matrix(aa)−1.Moreover, if we again define another non-zero vectord in VE , then it is easy to prove that thepermutation symmetries of elastic tensor indices (see equations (4.13)–(4.15)) assure that
(ad)αβ = (da)βα. (5.44)
We assume that a QC is an infinite, homogeneous, anisotropic and free body. Letm,n andtconstitute mutually orthogonal unit vectors inVE with t being parallel to a straight dislocationline. Letx denote the position vector from the origin to a point inVE . In the present case,uandw induced by infinite straight dislocation lines parallel tot are independent oft,x anddepend only on two orthogonal coordinatesy1 =m · x andy2 = n · x in the plane ofm andn in VE . Following Stroh, let us suppose a displacement field solution of the form
V β(x) = Aβf (λ) = Aβf (m · x + pn · x) (5.45)
whereAβ is a 6D constant complex vector,p is a complex constant and f is an analyticalfunction of its arguments. Using the expressions in equations (5.28)–(5.30), and substitutingequation (5.45) into (4.28), we obtain
{(mm)αβ + p[(mn)αβ + (nm)αβ ] + p2(nn)αβ}Aβ = 0. (5.46)
The set of equations (5.46) has non-zero solutions forAβ only if the determinant of thecoefficient matrix vanishes:
det{(mm)αβ + p[(mn)αβ + (nm)αβ ] + p2(nn)αβ} = 0. (5.47)
Since the above equation is a polynomial with real coefficients, the 12 rootspµ(µ =1, 2, . . . ,12) occur in complex-conjugate pairs. Supposepµ(µ = 1, 2, . . . ,6) equal theroots with positive imaginary parts, and thenpµ+6 = p∗µ. If we define another vectorLαµ as
Lαµ = −[(nm)αβ + pµ(nn)αβ ]Aβµ
or equivalently
Lαµ =1
pµ[(mm)αβ + pµ(mn)
αβ ]Aβµ (5.48)
the normalization condition should be satisfied
2AαµLαµ = 1 (5.49)
which ensure uniqueness forAαµ andLαµ . Furthermore, the following conjugate relations hold:
pµ+6 = p∗µ Aαµ+6 = (Aαµ)∗ Lαµ+6 = (Lαµ)∗ (µ = 1, 2 . . . ,6). (5.50)
It should be noted that the sum rule is no longer available for the repeated Greek subscriptin equations (5.48) and (5.49). From now on summation over repeated Greek subscripts willbe indicated explicitly. It is straightforward to extend Stroh’s 6D formalism (1962) to a 12Dformalism, i.e.
V β(x) = 1
2π i
12∑µ=1
±AβµLαµBα ln(m · x + pµn · x)
∂
∂xpV β(x) = 1
2π i
12∑µ=1
±AβµLαµBαmp + pµnp
m · x + pµn · x
(5.51)
where + is taken forµ = 1, 2, . . . ,6,− for µ = 7, 8, . . . ,12. By using the definition (5.48),equation (5.46) can be rewritten as
Nξµ = pµξµ
Physical properties of quasicrystals 33
with
ξµ =[Aµ
Lµ
]and
N = −[
(nn)−1(nm) (nn)−1
(mn)(nn)−1(nm)− (mm) (mm)(nn)−1
](5.52)
which is a 12D eigenvalue problem. Following the analogous derivation given by Baconet al(1978) and Barnett and Lothe (1975), one can prove that the orthogonality, completeness andinvariance are still valid forAαµ,L
αµ andpµ, i.e.
AαλLαµ +AαµL
αλ = δλµ
12∑µ=1
AαµLβµ = δαβ
12∑µ=1
AαµAβµ = 0
12∑µ=1
LαµLβµ = 0
∂
∂θAαµ = 0
∂
∂θLαµ = 0
(5.53)
whereθ is the angle between the plane basis(m,n) and a new plane basis(m′,n′) in theplane normal tot. Additionally,
∂pµ
∂θ= −(1 +p2
µ)∫ 2π
0pµ dθ = ±2π i.
(5.54)
Furthermore, we can define three 6×6 real matrices and give integral representations of them:
Qαβ = Qβα = i12∑µ=1
±AαµAβµ =−1
2π
∫ 2π
0[(nn)−1]αβ dθ
Bαβ = Bβα = −1
4π i
12∑µ=1
±LαµLαµ =1
8π2
∫ 2π
0{(mm)αβ − (mn)αγ [(nn)−1]γ λ(nm)λβ} dθ
Sαβ = i12∑µ=1
±AαµLβµ =−1
2π
∫ 2π
0[(nn)−1]αγ (nm)γβ dθ.
(5.55)
If we choosem along the position vectorx, thenm · x = |x| andn · x = 0. Therefore, in apolar coordinate system we have
mp∂
∂xp= ∂
∂rnp
∂
∂xp= 1
r
∂
∂θ(5.56)
wherer = |x|. Using equations (5.48) and (5.55), equation (5.51) can be expressed in termsof Sαβ andBαβ as
∂
∂xpV β(x) = bβ
2πr{−mpSβα + np[(nn)−1]βγ [4πBγα + (nm)γλSλα]}. (5.57)
Multiplying equation (5.57) bymp andnp, respectively, we have
∂
∂rV β(x) = − 1
2πrbαSβα
∂
∂θV β(x) = 1
2πbα[(nn)−1]βγ [4πBγα + (nm)γλSλα].
(5.58)
34 C Z Hu et al
Integrating this equation yields the representation of the displacement field
V β(r, θ) = 1
2πbα{− Sβα ln r + 4πBγα
∫ θ
0[(nn)−1]βγ dθ + Sλα
∫ θ
0[(nn)−1]βγ (nm)γλ dθ
}.
(5.59)
Meanwhile, inserting equation (5.57) into (5.38), the resulting stress field is
Mαj = Bαβjl
bϕ
2πr{−mlSβϕ + nl [(nn)
−1]βγ [4πBγϕ + (nm)γλSλϕ ]}. (5.60)
The elasticity theory of dislocations has been applied to QCs with some symmetries bya number of authors (Dinget al 1994, 1995b, Yanget al 1995a, Qinet al 1997, Li and Fan1998).
Acknowledgments
The authors were supported by a grant from the National Natural Science Foundation of China.Special thanks are given to Jianbo Wang and Zhendong Wang for technical help in preparingthe manuscript.
Appendix. Character analysis
It should be noted that there are two different coordinate transformations in the case of QCs,one associated with the physical spaceVE , and the other with the perpendicular spaceVI .From now on we use subscriptsi, j , k, l, . . . for coordinate components inVE and subscriptsα, β, γ , . . . for coordinate components inVI . Hence, the physical property tensors in QCsmay be classified as follows:
Fi, Fij , Fijk, Fijkl, . . . , Fαi, Fiαj , Fijβl, Fαiβj , . . . . (A.1)
Furthermore, some internal symmetries yield additional restrictions. It implies that somesubscripts are commutable. We shall use the notation{ } to represent the fact that the quantitiesin { } can commute with each other.
In the following list we give the various types of tensors which we may deal with in thecase of QCs. To simplify the writing we introduceA = 0A(g),B = 0B(g),χA(g) = Tr 0A(g)andχB(g) = Tr 0B(g).
(1) Second-rank tensors of typeF{ij}
[0(g)]ij,i ′j ′ = {A× A}ij,i ′j ′ = 12(Aii ′Ajj ′ +Aij ′Aji ′)
χ(g) = 12(AiiAjj +AijAji) = 1
2[χA(g)2 + χA(g
2)].(A.2)
(2) Second-rank tensors of typeFαi
[0(g)]αi,α′i ′ = (B × A)αi,α′i ′ = Bαα′Aii ′χ(g) = BααAii = χB(g)χA(g).
(A.3)
(3) Third-rank tensors of typeFi,{jk}
[0(g)]ijk,i ′j ′k′ = (A× {A× A})ijk,i ′j ′k′ = Aii ′ {A× A}jk,j ′k′= Aii ′ 12(Ajj ′Akk′ +Ajk′Akj ′)
χ(g) = Aii 12(AjjAkk +AjkAkj ) = 1
2χA(g)[χA(g)2 + χA(g
2)].(A.4)
Physical properties of quasicrystals 35
(4) Third-rank tensors of typeFi,αk
[0(g)]iαk,i ′α′k′ = (A× (B × A))iαk,i ′α′k′ = Aii ′(B × A)αk,α′k′ = Aii ′Bαα′Akk′χ(g) = AiiBααAkk = χB(g)χA(g)2. (A.5)
(5) Fourth-rank tensors of typeF{{ij},{kl}}
[0(g)]ijkl,i ′j ′k′l′ = {{A× A} × {A× A}}ijkl,i ′j ′k′l′= 1
8(Aii ′Ajj ′ +Aij ′Aji ′)(Akk′All′ +Akl′Alk′)+1
8(Aik′Ajl′ +Ail′Ajk′)(Aki ′Alj ′ +Akj ′Ali ′)
χ(g) = 18χA(g)
4 + 14χA(g)
2χA(g2) + 3
8χA(g2)
2+ 1
4χA(g4).
(A.6)
(6) Fourth-rank tensors of typeF{αi,βj}
[0(g)]αiβj,α′i ′β ′j ′ = {(B × A)× (B × A)}αiβj,α′i ′β ′j ′= 1
2[Bαα′Aii ′Bββ ′Ajj ′ +Bαβ ′Aij ′Bβα′Aji ′ ]χ(g) = 1
2[χB(g)2χA(g)
2 + χB(g2)χA(g
2)].(A.7)
(7) Fourth-rank tensors of typeF{ij},αk
[0(g)]ijαk,i ′j ′α′k′ = ({A× A} × (B × A))ijαk,i ′j ′α′k′= 1
2[Aii ′Ajj ′ +Aij ′Aji ′ ]Bαα′Akk′
χ(g) = 12[χA(g)
2 + χA(g2)]χB(g)χA(g).
(A.8)
Using the above formulae, one can easily find the character of a given tensor. For example,the thermal expansion coefficientα(1)ij is of typeF{ij} given in item (1). The tensor characterfollows from equation (A.2):
χ(e) = 6, χ(α) = 2 +√
2, χ(α2) = 0,
χ(α3) = 2−√
2, χ(α4) = χ(β) = χ(αβ) = 2(A.9)
wheree is the identity operation. From equation (3.16) the number of independent tensorcomponentsα(1)ij is
116[6 + 2(2 +
√2) + 2(2−
√2) + 2 + 4· 2 + 4 · 2] = 2. (A.10)
The thermal expansion coefficientα(2)ij is of typeFαi given in item (2). It follows fromequation (A.3) that
χ(e) = 6, χ(α) = −2−√
2, χ(α3) = −2 +√
2,χ(α4) = 2, χ(α2) = χ(β) = χ(αβ) = 0
(A.11)
and according to equation (3.16) the number of independent tensor componentsα(2)ij is
116[6− 2(2 +
√2) + 2(−2 +
√2) + 2] = 0. (A.12)
The piezoelectric constantsd(1)ijk andd(2)ijk are of typesFi,{jk} andFi,αk given in items (3) and(4), respectively. The tensor characters follow from equations (A.3) and (A.4):
χ(e) = 18, χ(α) = 4 + 3√
2, χ(α2) = 0, χ(α3) = 4− 3√
2,
χ(α4) = −2, χ(β) = χ(αβ) = 2
and
χ(e) = 18, χ(α) = −(4 + 3√
2), χ(α3) = −4 + 3√
2,χ(α4) = −2, χ(α2) = χ(β) = χ(αβ) = 0.
(A.13)
36 C Z Hu et al
The numbers of independent components are116[18 + 2(4 + 3
√2) + 2(4− 3
√2)− 2 + 4 · 2 + 4 · 2] = 3
for d(1)ijk and
116[18− 2(4 + 3
√2) + 2(−4 + 3
√2)− 2] = 0 (A.14)
for d(2)ijk , respectively. Similarly, the elastic constantsCijkl ,Kijkl andRijkl are of typesF{{ij},{kl}},F{αi,βj} andF{ij},αk in items (5), (6) and (7), respectively. We can derive, from equations (A.6)–(A.8), the tensor characters
χ(e) χ(α) χ(α2) χ(α3) χ(α4) χ(β) χ(αβ)
21 3 + 2√
2 1 3− 2√
2 5 5 5
21 3 + 2√
2 1 3− 2√
2 5 3 3
36 −6− 4√
2 0 −6 + 4√
2 4 0 0.
(A.15)
The numbersnC , nK , andnR of independent elastic constantsCijkl ,Kijkl andRijkl are
nC = 5, nK = 4, nR = 1. (A.16)
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