Wave propagation in micromorphic anisotropic continua with an application to tetragonal crystals
WWave propagation in micromorphic anisotropic continuawith an application to tetragonal crystals
Fabrizio Dav´ı DICEA and ICRYS, Universit´a Politecnica delle Marche,via Brecce Bianche, 60131 Ancona, Italy
Abstract
We study the coupled macroscopic and lattice wave propagation in anisotropiccrystals seen as continua with affine microstructure (or micromorphic). Inthe general case we obtain qualitative informations on the frequencies andthe dispersion realtions. These results are then specialized to crystals of thetetragonal point group for various propagation directions: exact represen-tation for the acoustic and optic frequencies and for the coupled vibrationsmodes are obtained for propagation directions along the tetragonal c -axis. Keywords:
Anisotropic crystals, Lattice vibrations, Acoustic waves, Opticwaves, Scintillating crystals.
1. Introduction
Let me say from the beginning that this paper was motivated by secondtoughts: indeed my research deals mostly with the photoelastic properties ofscintillating crystals, that is crystals which convert ionizing radiations intophotons within the visible range. Massive scintillating crystals were used todetect particle collisions in the CMS calorimeter at CERN, Geneve [1] andshall be used in the FAIR accelerator at GSI, Darmstadt [2] and can also beused into security and medical imaging devices. Amongst many other prob-lems concerning quality control and efficency, one of the major issues relatedwith the prolongated use of these crystals is the radiation damage which dis- e-mail: [email protected] Preprint submitted to Elsevier September 22, 2020 a r X i v : . [ phy s i c s . c l a ss - ph ] S e p laces atoms and reduces crystal efficiency and the radiation/photons ratio( vid. e.g. [3]).In order to recover the radiation damage many techniques were used:between them one of the most promising is the laser-induced ultrasoundlattice vibration, which can be a really efficient way to recover the damage. Itbecomes mandatory, therefore, to study the problem of coupled bulk/latticewaves propagation in crystals in order to evaluate the frequency range of bulkand lattice vibrations, the amount of energy which is lost in the couplingbetween lattice and bulk and how much of the incoming energy makes thelattice to vibrate and around which modes. The problem is remarkablycomplex since most scintillators exhibit strong anisotropy, like the monoclinicCerium doped Lu x Y − x SiO :Ce (LYSO) [4], the hexagonal LaBr [5] and thetetragonal PbWO (PWO) [6].Such problem can of course be studied with a classical lattice dynamicsapproach [7], [8]: however, since scintillators generally are massive crystalwhose size is in the range of decimeters, the continuum mechanics approachlooks more suited to describe the interactions between the crystal lattice vi-brations and the macroscopic vibrations of the crystal specimen. Thus itseem natural to model the crystal as a continuum with affine structure [9]or micromorphic continuum [10], since it appears as a reasonable compro-mise between the microscopical aspects related to lattice vibrations and themacroscopic vibrations.Micromorphic continua have attracted a never-fading attention since thepionieering work of Mindlin [11], the treatises [9] and [10] and the revampedattention in the recent years, motivated by the study of metamaterials, bythe means of both the classical approach as in [12] or the relaxed one firstproposed into [13]. The majority of these results however concern isotropicmaterials, with some limited exceptions. Here for classical micromorphiccontinua, we extend to the general case of anisotropic materials the previouslyknown results of wave propagation into isotropic material, and then specializethem to crystals of the tetragonal group.The paper is organized as follows: in § .2 we write the balance law asproposed into [9] and then, by using the results of [14] we show that theyare fully equivalent to those given in [11]; upon the assumption of linearizedkinematics and by using linear constitutive relations as in [11], we arriveat the propagation condition for the macroscopic progressive waves coupledwith microdistortions lattice waves. Such propagation condition, which de-pends on the wavenumber ξ and on two dimensionless parameters which2elates the various length-scales of the problem, is completely described by a12 ×
12 Hermitian matrix whose blocks represents various kind of generalizedacoustic tensors and whose eigenvector represents macroscopic displacementscoupled with lattice microdistortions. In the general case of triclinic crystalswe show that there ever exist three acoustic and nine optic waves and wealso give an insight into the structure of dispersion relations: moreover bya suitable scaling in terms of the dimensionless parameters we show thatthe problem admits two physically meaningful limit cases, namely the
Longwavelength approximation which represents the propagation phenomena ina body in which we are ”zooming-out” away from the crystal lattice, andthe
Microvibration case, where on the converse we are ”zooming-in” intothe crystal lattice. These two limit cases first introduced into [11], besidesrepresenting two physical picture of the phenomena, give an insight into thegeneral propagation problem with the cut-off optical frequencies given by themicrovibration frequencies.In the following § .3 these general results are specialized to crystals of theTetragonal point group and the reason for such a choice is two-fold: firstof all we are interested to damage recovery in the tetragonal PWO crystalswhich are currently used in the FAIR accelerator [15]; second, the reductionof independent constitutive parameter for tetragonal micromorphic bodiesallow for some explicit solutions of the propagation condition and to a atleast qualitative representation of the dispersion relations. We study sepa-rately the low-symmetric tetragonal classes 4, ¯4, 4 /m and the high-symmetricclasses 4 mm , 422, ¯42 m and 4 /mm , in the case of waves propagating along thetetragonal c -axis. For the other relevant case, that of waves propagating indirections orthogonal to the c -axis we give an insight to the solution structuresince along such directions the crystal behaves in a fully coupled manner as intriclinic crystal: hence it is not possible to give exact closed-form solutions.It there exists a major criticism to such an approach: as it was correctlypointed out into [16], such a formal treatment has some limitations since itdepends indeed on a large number of parameters whose experimental iden-tification can be both difficult and elusive. With respect to such a correctcriticism we can say that in any case we have a general framework to designcorrect experiments aimed to parameter identification and moreover, as it isshow into two recent papers [17], [18], by homogenization techniques we canestimate the micromorphic model constitutive parameters by the means ofclassical lattice dynamics.In the final § .4, the results for the full propagation condition and for the3wo limiting cases are given in tabular form; as far as we know this is the mostcomplete analysis of wave propagation in micromorphic tetragonal crystalsup to now and it could be the starting point for both experiment design andhomogenization techniques based on lattice dynamics. Let V be the three-dimensional vector space whose elements we denote v ∈ V and let Lin be the space of the second order tensors A ∈ Lin, A : V → V . We denote A T the transpose of A such that Au · v = A T v · u , ∀ u , v ∈ V ; we shall also denote Sym and Skw the subspaces of Lin of thesymmetric ( A = A T ) and skew-symmetric ( A = − A T ) tensors respectively.Let Lin be the space of third-order tensors P : Lin → V and for all P ∈ Lin we denote the transpose P T : V →
Lin as: P [ A ] · v = P T v · A , ∀ v ∈ V , ∀ A ∈ Lin . (1)We shall also make use of fourth-order tensors C : Lin → Lin, fifth-order tensors F : Lin → Lin and sixth-order tensors H : Lin → Lin whosetranspose are defined by the means of C [ A ] · B = C T [ B ] · A , ∀ A , B ∈ Lin , F [ A ] · P = F T [ P ] · A , ∀ A ∈ Lin , ∀ P ∈ Lin , (2) H [ P ] · Q = H T [ Q ] · P , ∀ P , Q ∈ Lin . For { e k } , k = 1 , , V , we define the componentsof the aforementioned elements by: v k = v · e k ,A kj = Ae j · e k = A · e k ⊗ e j , P ihk = P [ e h ⊗ e k ] · e i = P · e i ⊗ e h ⊗ e k , C ijhk = C [ e h ⊗ e k ] · e i ⊗ e j , i, j, h, k, m, p = 1 , , , (3) F ijhkm = F [ e h ⊗ e k ⊗ e m ] · e i ⊗ e j , H ijkhmp = H [ e h ⊗ e m ⊗ e p ] · [ e i ⊗ e j ⊗ e k ] . We shall also made use of the orthonormal base { W k } , k = 1 , . . . , W = e ⊗ e W = e ⊗ e W = e ⊗ e , W = e ⊗ e W = e ⊗ e W = e ⊗ e , (4) W = e ⊗ e W = e ⊗ e W = e ⊗ e , { ˆ W k } , k = 1 , . . . , { ¯ W k } , k =4 , , W k = W k , k = 1 , , , (5)ˆ W = ( W + W ) , ˆ W = ( W + W ) , ˆ W = ( W + W ) , and¯ W = 12 ( W − W ) , ¯ W = 12 ( W − W ) , ¯ W = 12 ( W − W ) . (6)In terms of the bases (4)-(6) we can also represent the fourth-order tensorscomponents with the so-called Voigt two-index notation, namely e.g. for (4): C ij = C [ W j ] · W i , i, j = 1 , . . . , . (7)Finally, in order to describe the infinitesimal lattice vibrations we shallmake use of the following seven modes: • Non uniform dilatation: D = α W + β W + γ W ; (8) • Dilatation along e and uniform plane strain in the plane orthogonalto e : D = α ( I − W ) + γ W ; (9) • Traceless plane strain orthogonal to e : D = W − W ; (10) • Shear in the plane orthogonal to e : S = α ˆ W ; (11) • Shear between e and the direction e ⊥ = α e + β e : S = − α ˆ W + β ˆ W ; (12) • Rigid rotation around the direction e : R = ω ¯ W ; (13) • Rigid rotation around the direction e ⊥ = ω e + ω e : R = ω ¯ W − ω ¯ W . (14)5 . Crystal as a micromorphic continuum Let B a region of the Euclidean three-dimensional space we pointwiseidentify with the reference configuration of a crystal, and let x be a point of B . We assume that at each point x ∈ B is defined a crystal lattice { a , a , a } with a × a · a ≥ x ∈ B and at each time t ∈ [0 , τ )it is well-defined the motion by the means of the two fields: y = y ( x , t ) , G = G ( x , t ) (15)provided y is locally injective and orientation-preserving and G is orientation-preserving, namely:det F > , F ( x , t ) = ∇ y ( x , t ) , det G > , (16)in such a way that at at each point y ∈ B t ≡ y ( B , t ) the deformed crystallattice { ¯ a , ¯ a , ¯ a } is given by:¯ a k = Ga k , k = 1 , , . (17)We identify B , endowed with the motion (15), with a continuum withaffine structure [9] or micromorphic [19], whose underlying manifold is M =Lin + and whose balance laws are given by: • the balance of macroforces:div T T + b = ρ ˙ v , (18)where T is the (non-symmetric) Cauchy stress, b is the volume forcedensity, ρ is the mass density and v the material velocity; • the balance of microforces:div T − K + B = ρ ¨ GJ , (19)where the third-order tensor T represents the microstress, K the in-teractive microforce, B the density of volume micro forces and J themicroinertia tensor 6 the balance of couples:skw( − T + GK T + (grad G ) T T ) = . (20)A different set of balance laws was provided in [11]: in order to recoverthese balance laws from (18)-(20) first of all we notice that skw T = − skw T T and set: ¯ T = T T + GK T + (grad G ) T T ∈ Sym ; (21)then, by following [11], we define the relative stress as: S = − ( GK T + (grad G ) T T ) , (22)in such a way that T T = ¯ T + S . (23)As second step we multiply the transpose of (19) for G to obtain: G (div T ) T − GK T + GB T = ρ G ( ¨ GJ ) T , (24)and since G (div T ) T = div( G T T ) − (grad G ) T T , (25)then with the aid of (23) from (18), (24) and (25) we recover equations (4.1)of [11]: div( ¯ T + S ) + b = ρ ˙ v , (26)div H + S + ¯ B = ρ GJ ¨ G T , where H = G T T , ¯ B = GB T . (27) As it is shown in [9], [19], [10], there are many appropriate kinematicalmeasures for a constitutive theory of micromorphic continua; here we choosethose proposed into [10], eqn. (1.5.11): E = 12 ( F T F − I ) , M = F T G − − I , G = G − grad G , (28)where E is the Green-Lagrange deformation measure.7e assume that both the deformation gradient and the lattice deforma-tion can be decomposed additively into: F = I + ∇ u , G = I + L , (29)where u ( x ) = y ( x ) − x is the displacement vector and L is the microdisplace-ment or microdistortion [16]. If we assume that ε = sup {(cid:107)∇ u (cid:107) , (cid:107) L (cid:107)} , (30)then the kinematical measures (28) can be rewritten, to within higher-orderterms into ε , as: E = sym ∇ u , M = ∇ u T − L , G = ∇ L ; (31)the tensor M being called the relative strain [11] or relative distortion [16]. Remark 1.
By using (29) and (30) into (22) and (27) we have: H = T + O ( ε ) , S = − K T + O ( ε ) , ¯ B = B T + O ( ε ) , (32) and: GJ ¨ G T = J ¨ L T + O ( ε ) . (33) Accordingly, from the balance law (26) we obtain, to within higher-orderterms, the balance of microforces (19) with ¨ L in place of ¨ G : div T − K + B = ρ ¨ LJ , (34) whereas the macroforces balance (18) can be rewritten as div( ¯ T − K T ) + b = ρ ¨ u . (35) In the sequel we shall use H , S and ¯ T as the mechanical descriptors associatedto a linearized kinematics, since in this case they are equivalent to T , K and T . We assume a linear dependence of ¯ T , S and H on the linearized kinemat-ical variables (28) and write ( cf. eqn. (5.3) of [11]):¯ T = C [ E ] + D [ M ] + L c F [ G ] , S = D T [ E ] + B [ M ] + L c G [ G ] , (36) H = L c F T [ E ] + L c G T [ M ] + L c H [ G ] , where: 8 C : Sym → Sym, C = C T is the fourth-order elasticity tensor, whosecomponents obey: C ijhk = C jihk = C ijkh = C hkij , (37)and there are at most 21 independent components. • The fourth-order tensor B : Lin → Lin, B = B T , whose independentcomponents are at most 45: B ijhk = B hkij . (38) • The fourth-order tensor D : Lin → Sym has 54 independent compo-nents: D ijhk = D jihk , ( D T ) ijhk = D hkij . (39) • The fifth-order tensors F : Lin → Sym and G : Lin → Lin haverespectively 162 and 243 components which obey: F ijhkm = F jihkm , ( F T ) ijhkm = F hkmij , ( G T ) ijhkm = G hkmij . (40) • The sixth-order tensor H = H T has at most 378 independent compo-nents H ijkhmn which obey: H ijkhmn = H hmnijk . (41) • L c > correlation length which makes all thesetensorial quantities of the dimension of a stress (Force/Area) .The correlation length L c is the first length scale we need to introduce intothe model and is related to the non-local effects associated with the gradientof the microdistorsion tensor. For L c → L c → ∞ acts as a zoom into the microstruc-ture ( cf. [16]): we shall made these statements more rigorous in the nextsubsection.As in [11], the requirement that the energy density is positive2 E ( E , M , G ) = ¯ T · E + S · M + H · G > , (42)9mplies that C , B and H be positive definite: C [ A ] · A > , B [ A ] · A > , ∀ A ∈ Lin / { } , (43) H [ A ] · A > , ∀ A ∈ Lin / { } , as well as det( B − D T C − D ) > , det( C − DB − D T ) > , (44)det( H − F T C − F ) > , det( H − G T B − G ) > , and det C D L c F D T B L c G L c F T L c G T L c H > . (45)In the most general case, that of crystals of the Triclinic group, theseconstitutive relations require the knowledge of 903 material constants, sub-ject to the restrictions (43)-(45), whereas in the simplest case of Isotropicmaterials these constants reduce to 18 independent at most [11]. In the nextsubsection we shall give a general and formal treatment of waves propagationin a crystal without any of the restrictions given by crystal symmetries. The balance laws (26) written in terms of the linearized kinematics (31)by the means of the constitutive relations (36) and zero volume macro- andmicroforces: div( ¯ T + S ) = ρ ¨ u , (46)div H + S = ρ J [ ¨ L ] , where J [ ¨ L ] = J ¨ L T , J ijhk = δ ih J jk , (47)are the starting point for the description of the microscopic crystal latticevibrations coupled with the macroscopic bulk vibrations. We seek for (46)progressive plane wave solutions of the form: u ( x , t ) = a e iσ , L ( x , t ) = C e iσ , σ = ξ x · m − ωt , (48)10here ω is the frequency, m the direction of propagation, ξ = λ − > λ the wavelength and where a ∈ V and C ∈ Lin denoterespectively the displacement and microdistortion amplitudes which in thegeneral case are complex-valued.We find at this point mandatory to introduce, besides the characteristiclenght scale L c , two further length scales: the macroscopic length L m > a = L m a o , (49)with a o a dimensionless vector, and the lattice length L l > J = L l J o , (50)with J o a dimensionless fourth-order tensor [16]. If we define the two dimen-sionless parameters ζ = L c L m , ζ = L l L m , (51)then for ζ → ζ → ∞ we are zooming into the microstructure. In the case ζ → ∇ u = iξL m a o ⊗ m e iσ , ∇ L = iξ C ⊗ m e iσ , (52)¨ u = − ω L m a o e iσ , ¨ L = − ω C e iσ , then we are led, by (31), (36), (46), (51) and (52), to the propagation con-ditions written in terms of the two lengths λ = ξ − , L m and of the twodimensionless parameters ζ , ζ : ξ A ( m ) a o + ( ξ ζ P ( m ) + iξL − m Q ( m ))[ C ] = ω a o , (53)( ξ ζ P T ( m ) − iξL − m Q T ( m )) a + ( ξ ζ A ( m ) + L − m ¯ B )[ C ] = ω ζ J o [ C ] , A ( m ) a = ρ − ( C [ a ⊗ m ] + D [ m ⊗ a ] + D T [ a ⊗ m ] + B [ m ⊗ a ]) , P ( m )[ C ] = ρ − ( F + G )[ C ⊗ m ] m , Q ( m )[ C ] = ρ − ( D + B )[ C ] m , (54) A ( m )[ C ] = ρ − H [ C ⊗ m ] m , ¯ B = ρ − B , and whose representation in components are A ij = ρ − ( C iljk m k m l + D ilhj m h m l + D iljk m k m l + B ilhj m h m l ) , P ihk = ρ − ( F ijhkp + G ijhkp ) m p m j , Q ijh = ρ − ( D ijhk + B ijhk ) m k , (55) A ijhk = ρ − H ijlhkp m l m p , ¯ B ijhk = ρ − B ijhk . We notice that, since C = C T , B = B T and H = H T , from (55) we have that: A ∈ Sym , A = A T ; (56)we call A ( m ) the generalized acoustic tensor and A ( m ) the microacoustictensor.We define the two 12 ×
12 hermitian block matrix A ( ξ ) = A ∗ ( ξ ), whichwe call the Acoustic matrix , the 12 ×
12 symmetric block matrix J = J T andthe 12 dimension eigenvector w : A ( ξ ) ≡ ξ A ξ ζ P + iξL − m Q ξ ζ P T − iξL − m Q T ξ ζ A + L − m ¯ B , J ≡ I 00 ζ J o , w ≡ a o C , (57)in such a way that we can rewrite the propagation condition (53) as( A − ω J ) w = 0 . (58) Indeed in absence of the microstructure (54) reduces to the acoustic tensor for linearlyelastic bodies, vid. e.g. [21], § .70.
12e require, that A be positive-definite for all ξ > ξ →
0) and then, since A and B are positive-definite by (43) and the definition (54) , the positive-definiteness of A impliesthat also A ( m ) be positive definite; accordingly the eigenvalue problem (58)admits the twelve eigencouples with real eigenvalues ( ω k , w k ) , < J w h , w k > = δ hk , h, k = 1 , . . . , . (59)The characteristic equation associated with the propagation condition(58) is det( A ( ξ ) − ω J ) = 0 , (60)and since the components of M are functions of the wavenumber ξ , then alsothe eigencouples are: ξ (cid:55)→ ( ω k ( ξ ) , w k ( ξ )) , k = 1 , . . . , . (61)The functional dependences between ω and ξ are called the dispersion rela-tions and in terms of these relations we can define the phase v pk and groupvelocity v gk as: v pk ( ξ ) = ω k ( ξ ) ξ , v gk ( ξ ) = d ω k ( ξ )d ξ , k = 1 , . . . , . (62)As pointed out into [22], waves in micromorphic continua can be classifiedinto: • Acoustic waves , whose frequencies ω k ( ξ ) goes to zero for ξ → • Optic waves for which the limit for ξ → ω k (0) is called the cut-off frequency with group velocities v gk (0) =0 ; • Standing waves those associated to immaginary values ξ = ± ik , k > u ( x , t ) = a e ∓ k x · m e − iωt , L ( x , t ) = C e ∓ k x · m e − iωt ; (63) With the notation < a , b > = m (cid:88) h =1 a h b ∗ h , we denote the euclidean inner product on a m -dimensional complex space C m . ξ → A are analytic functions of ξ ≥ ξ = ξ o where the eigenvalues ω k ( ξ ) are regularand where their derivatives are well-defined. We use this result, in the case ofsimple eigenvalues, to give an explicit formula for the group velocities ([23],Thm. 5): v gk ( ξ o ) = 12 ω k ( ξ o ) < d A d ξ (cid:12)(cid:12)(cid:12) ξ = ξ o w k ( ξ o ) , w k ( ξ o ) > ; (64)moreover the eigenvectors w k ( ξ ) are differentiable functions of the wavenum-ber.Now, since for ξ → A is real with three multiple eigenvalues ω = 0, nine non-zero (simple) eigenvalues ˆ ω j and twelve real eigenvectors ˆ w k ,then we can use Theorem 2 of [23] to show that (60) admits three and onlythree zero eigenvalues as ξ approaches zero: moreover by the differentiabilityof eigenvalues and eigenvectors we have that:ˆ ω k = lim ξ → ω k ( ξ ) , ˆ w k = lim ξ → w k ( ξ ) , (65)and for the non zero eigenvalues we have from (64) v gj (0) = 12ˆ ω j < d A d ξ (cid:12)(cid:12)(cid:12) ξ =0 ˆ w j , ˆ w j > = 0 . (66)Therefore, by the application of the results obtained in [23] to our case, weobtain that in any anisotropic crystal : • There exist three Acoustic waves and nine Optic waves; • The cut-off frequencies for the Optic waves are the limit as ξ → • The frequencies for the Acoustic waves are the eigenvalues of (58) whichgoes to zero in the limit for ξ → • The eigenvectors for ξ → A would be not positive-definite and therefore we canconclude that no standing waves are possible within this anisotropic model.Indeed as it was observed into [13], [24], [25] and [22] for the isotropic case,standing waves are associated with band-gap material and are not possiblewithin the classical micromorphic model: they appears instead in the relaxedmicromorphic model proposed into [13].The solutions of the eigenvalues problem (58) depend, besides the param-eter ξ , also on the three parameters L m , ζ and ζ whose limiting values, aswe already remarked, corresponds to different physical scales: therefore, be-sides the complete condition given by (53), we shall study into some detailsthese two limit cases. If we let ζ → ζ → long wavelength approximation [16]. The propagation conditions(53) then reduce, to within higher-order terms in ζ and ζ , to: ξ A ( m ) a o + iξL − m Q ( m ))[ C ] = ω a o , (67) − iξ Q T ( m ) a o + L − m ¯ B [ C ] = ;since B is positive definite, then from (67) we have: C = iL m ξ B − Q T ( m ) a o , (68)and therefore from (67) we obtain the classical continuum propagation con-dition ˆ A ( m ) a o = c a o , ω = cξ , c = v p = v g , (69)where the acoustic tensor ˆ A ( m ) (which is independent on the macroscopiclength L m ) is defined as:ˆ A ( m ) a o = A ( m ) a o − Q ( m ) B − Q T ( m ) a o = ( C − C micro )[ a o ⊗ m ] m . (70)In this definition C is the elasticity tensor from (36), whereas by using (54) we can represent the positive-definite microelasticity tensor C micro as: C micro = DB − D T , C micro = C Tmicro , C micro : Sym → Sym ; (71)15y (44) C − C micro is positive-definite and hence the acoustic tensor ˆ A ( m )is positive-definite too.The three eigencouples of (69) represents acoustic waves; however in thisapproximation ˆ A ( m ) is not the acoustic tensor of the linear elasticity andthe presence of the microstructure makes the propagation velocities smallerthan in linearly elastic bodies: moreover we have also three microdistortionsassociated to the eigenvectors of (69) by the means of (68); these microdistor-sions are purely immaginary and depend on the ratio L m ξ = L m /λ betweenthe macroscopic scale and the wavelength. L c → : microvibrations If we let L c → ∞ , for fixed L m ≈ L l , then we are zooming into the crystal;into the limit the constitutive relations (36) , remain finite for any choice ofmaterial only if ∇ L = . In this case, from (48) we have that ξ = 0 and thepropagation condition (53) leads to a solution ω = 0 with multiplicity three.Since by (48) u reduces to a rigid motion, then without loss of generalitywe can set: u ( x , t ) = , L ( t ) = C e iωt , (72)and from the propagation condition (53) then we are led to the characteristicequation det( B − ρω J ) = . (73)For L c → ∞ we therefore recover the Microvibration solutions of (53), whichwas studied in detail into [11] for isotropic materials ( vid. also [22]); thepropagation condition (73) admits nine eigencouples ( ω k , C k ), k = 1 , . . . , § . 2.3.
3. Wave propagation in Tetragonal crystals
As we already remarked an explicit solution for the propagation condition(58) and an explicit determination for the dispersion relations (61) is notpossible in the general anisotropic case, nor it would be particularly useful,since the associated kinematics would be fully coupled. However many of thecomponents of both the acoustic tensors A and A , as well as of P and Q mayvanish according to both the crystal symmetry group and the propagation For the solution of (69) for the various symmetry groups one can refer to [26] or [27]. m and it could make sense to obtain explicit solutions for specialcases of symmetry and directions of propagation.The simplest case of isotropic material (which depends only on 18 consti-tutive parameters) was studied in full-length into [11] and further analizedand extended to a relaxed micromorphic model into [13], [20]). In this sectionwe shall study the wave propagation condition (58) for crystals belonging tothe Tetragonal symmetry group. We took e directed as the tetragonal c -axis, hence the acoustic tensorshave the representation given into the § .5.4 of the Appendix. At a glance,by looking at (158), (159), (162), (164), (166), (170), (173) and (174) for ageneric propagation direction m , the matrix A does not simplify enough andthe problem maintains the same complexity as for crystal of the Triclinicgroup.However, for the two relevant cases of propagation direction either par-allel ( m × e = ) or orthogonal ( m · e = 0) to the tetragonal c -axis, manyof the components of A vanish and the number of the independent compo-nents reduces too, allowing for an explicit solution of (58) whose associatedkinematics can be understood more easily. Therefore we study these twopropagation direction and we begin with the two limit propagation prob-lems we obtained into § . 2.3.1 (long-wavelength approximation) and § . 2.3.2(microvibrations). For the propagation direction m = e ( i.e. along the tetragonal c − axis:henceforth we shall use c and e as synonimus when we describe the mate-rial symmetry) the tensor (160) reduces for all classes to the isotropic-likerepresentation: ˆ A ( e ) = 1 ρ (cid:16) ˆ C ( I − W ) + ˆ C W (cid:17) , (74) The propagation problme in Tetragonal materials was previuosly studied into [28]:however their analysis, which concerns a relaxed micromorphic model rather then theclassical one, was limited to two-dimensional plane strain; their main focus was indeedthe parameter identification by means of numerical homogeneization (see the commentsat the end of § .4). ω ( ξ ) = ω ( ξ ) = ξ (cid:115) ˆ C ρ , a = cos β e + sin β e , a = − sin β e + cos β e , (75) ω ( ξ ) = ξ (cid:115) ˆ C ρ , a = e , By (68), these macroscopic displacements are accompained by a microdis-tortion associated with the longitudinal wave of frequency (75) C = iL m ξ B − Q T ( e ) e ; (76)since the tensor Q ( e ) for the classes 4, ¯4 and 4 /m has the tabular representa-tion (163) with the non-null components given by (162) and since B − has thesame non-null components of (147), then the tensor C can be representedas C = α ( I − W ) + β W + γ ¯ W , (77)where: α = Q ( B − + B − ) + Q B − + Q ( B − − B − ) ,β = Q B − + 2 Q B − + 2 Q B − , (78) γ = Q ( B − + B − ) + Q B − + Q ( B − − B − ) , and the components Q , Q and Q are obtained from (162) evalutedfor m = m = 0 and m = 1.The microdistortions associated to the longitudinal waves are therefore acombination of the modes D and R .The microdistortions accompanied to the transverse waves which propa-gates along m = e are instead given by C k = iL m ξ B − Q T ( e ) a k , k = 1 , , (79)which can be represented as C k = α k ˆ W + β k ˆ W + γ k ¯ W + δ k ¯ W , k = 1 , , (80)18ith, for instance2 α = ( Q C − Q S )( B − + B − ) + ( Q C − Q S )( B − + B − )+ B − (( Q − Q ) C − ( Q + Q ) S ) , β = ( Q C − Q C )( B − + B − ) + ( Q C + Q S )( B − + B − )+ B − (( Q + Q ) C − ( Q + Q ) S ) , γ = ( Q C − Q C )( B − − B − ) + ( Q C + Q S )( B − − B − )+ B − (( Q − Q ) C + ( Q − Q ) S ) , δ = ( Q C − Q S )( B − − B − ) + ( Q C − Q S )( B − − B − )+ B − (( Q + Q ) C − ( Q − Q ) S ) , where C = cos β , S = sin β and with Q , Q , Q and Q given by (162)with m = 1 and m = m = 0.Therefore each transverse wave a k which propagates along m = e gen-erates a combination of the modes S and R . Remark 2 (Classes mmm , , /mm , ¯42 m ). For these classes B − hasthe same non-null components of (147) and we also have Q = Q = Q = 0 . The only noticeable difference with the results thus far obtained isthat γ = 0 into (78) and hence C reduces to the mode D . Whenever the propagation direction is orthogonal to the c − axis, say m =cos θ e + sin θ e then we have, for all θ , a transverse wave which is directedas the c − axis: ω ( ξ ) = ξ (cid:115) ˆ C ρ , a = e (81)and two waves which in general are neither transverse nor longitudinal: ω , ( ξ ) = ξ (cid:114) ρ (cid:16) a ± (cid:112) b cos θ + c sin θ + 2 d sin 2 θ cos 2 θ (cid:17) , a = cos β e + sin β e , (82) a = − sin β e + cos β e , with tan β = a − √ b cos θ + c sin θ + 2 d sin 2 θ cos 2 θ C cos 2 θ + ( ˆ C + 2 ˆ C ) sin 2 θ , (83)19here, for the classes 4, ¯4 and 4 /m : a = ˆ C + ˆ C , b = ( ˆ C − ˆ C ) + 4 ˆ C , (84) c = ˆ C + ( ˆ C + 2 ˆ C ) , d = ˆ C ( ˆ C + ˆ C + 4 ˆ C ) . We may search for the angle θ such that a is a longitudinal wave withfrequency ω and a is a transverse wave with frequency ω , which is equiv-alent to require either that a × m = and hence θ = β , or that A = 0which gives tan 2 θ = − C ˆ C + 2 ˆ C . (85)For the classes 4 mm , , /mm and ¯42 m with ˆ C = 0 then b = ( ˆ C − ˆ C ) , c = ( ˆ C + 2 ˆ C ) , d = 0 , (86)and then from (85) and (83) with β = θ it is easy to see that for θ ∈{ , π/ , π/ } we have a longitudinal ( a = m ) and a transverse ( a = e × m )waves whose frequencies are given by (82) when we use (86) in place of (84).The microdistortion which corresponds to the transverse wave (81) is: C = iL m ξ B − Q T ( θ ) e , (87)and by (163) and (147) we obtain again (80) with for instance, when k = 1:2 α = Q ( B − + B − ) + Q ( B − − B − ) + Q B − + Q B − , β = Q ( B − + B − ) + Q ( B − + B − ) − Q B − + Q B − , (88)2 γ = Q ( B − − B − ) + Q ( B − + B − ) + Q B − − Q B − , δ = Q ( B − − B − ) + Q ( B − − B − ) − Q B − − Q B − ;here the components Q α ( θ ) and Q α ( θ ), α = 1 , m = cos θ , m = sin θ and m = 0.Accordingly this transverse wave is associated to a shear between m and e and a rigid rotation about e × m .When we turn our attention to the other two waves, which are neithertransverse nor longitudinal, we have that the two associated microdistortions C γ = iL m ξ B − Q T ( θ ) a γ , γ = 2 , , (89)20n view of (147) and (163) can be represented as: C = iL m ξ (cos β B + sin β B ) , (90) C = iL m ξ ( − sin β B + cos β B ) ;the two tensors B k = B − Q T ( θ ) e k , k = 1 , B k = α k W + β k W + γ k W + δ k ˆ W + (cid:15) k ¯ W , (91)where α k = B − Q k + B − Q k + B − Q k + B − Q k + B − Q k ,β k = B − Q k + B − Q k + B − Q k + B − Q k + B − Q k ,γ k = B − ( Q k + Q k ) + B − Q k + B − ( Q k − Q k ) , k = 1 , , (92) δ k = 12 ( B − − B − )( Q k + Q k ) + 12 ( B − − B − )( Q k + Q k ) ,(cid:15) k = 12 ( B − + B − )( Q k + Q k ) + 12 ( B − − B − )( Q k − Q k ) + B − Q k ;here the components of Q ( θ ) are obtained again from (162) with m = cos θ , m = sin θ and m = 0.Interestingly enough for both the waves with amplitudes a and a , thecorresponding microdistortions are a combination of the modes D , S and R . We remark that this situation is maintained even when the propagationdirection is given by (85) and the two waves becomes one transverse and theother longitudinal. Remark 3 (Classes mmm , , /mm , ¯42 m ). For these classes, by (164)we have that formula (91) holds with α k = β k and δ k = (cid:15) k and therefore themode D changes into D .3.2. Microvibrations3.2.1. Classes , ¯4 and /m We begin with the lower-symmetry tetragonal classes: when we considerthe propagation condition (73) with B and J given respectively by (147) and(150), we notice that both tensors are reduced by the two subspaces of Lin, U and U with Lin ≡ U ⊕ U : U ≡ span { W , W , W , W , W } , (93) U ≡ span { W , W , W , W } , B [ C α ] ∈ U α , J [ C α ] ∈ U α , ∀ C α ∈ U α , α = 1 , . (94)The eigencouples split accordingly into two group: a first one ( ω k , C k ), k =1 , . . . C k ∈ U and whose eigentensors are a combination of the modes D k , S and R ; the second one ( ω j , C j ), j = 6 , . . . C j ∈ U and whoseeigentensors are a combination of the modes S and R .We define the normalized components of B as follow: a = B ρJ , b = B ρJ , d = B ρ √ J J , e = B ρ √ J J ,c = B ρJ , f = B ρJ , g = B ρJ , h = B ρJ , l = B ρ √ J J , (95) m = B ρJ , n = B ρJ , p = B ρ √ J J , q = B ρ √ J J , and we begin our analysis with the subspace U . We notice that the algebraicfifth-order characteristic equation in ω can be factorized into( ω − Bω + C )( ω + Dω + Eω + F ) = 0 , (96)where B = a + c + f − d ,C = ( a − d )( c + f ) − ( g − h ) ,D = a + b + c + d − f , (97) E = 2 a ( b + c ) + bc − f ( a + b + d ) − e + l ) − ( g + h ) ,F = ( c − b ) − ( g + h ) − d − e + f + 18 l , + 3 d ( b + c ) − f (2 b + c + d ) − gh − bc . The eigenvalues are thus given by ω , = 12 ( B ∓ √ B − C ) , (98)with ω < ω and, by the means of Cardano’s formulae [29], by: ω = 13 ( D + 2 √ P cos θ ,ω = 13 ( D + 2 √ P cos θ + 2 π , (99) ω = 13 ( D + 2 √ P cos θ − π , ω < ω < ω or ω > ω > ω and where: P = D − E , Q = D − DE − F , θ = cos − Q √ P . (100)The associated eigentensors combine the modes D , S , R in all but onecase, when it combines D and R : C = α W + β W + γ W + δ ˆ W + (cid:15) ¯ W , C = β W + α W + γ W − δ ˆ W + (cid:15) ¯ W , C = α ( I − W ) + γ W + 2 δ ¯ W , (101) C = α W + β W + γ W + δ ˆ W + (cid:15) ¯ W , C = β W + α W + γ W − δ ˆ W + (cid:15) ¯ W ;here the real coefficients α k , β k , γ k , δ k and (cid:15) k , which depend on the compo-nents of B , are given explicitly into the Appendix, § .5.5.1.When we turn our attention to the subspace U , then we get two eigen-values of multiplicity 2: ω = ω = 12 ( m + n ) − (cid:114) ( m − n + p + q , (102) ω = ω = 12 ( m + n ) + (cid:114) ( m − n + p + q , whose (non-normalized ) eigentensors are C = p ˆ W − p ¯ W + ( n − q − ω ) ˆ W − ( q + n − ω ) ¯ W , C = ( m − q − ω ) ˆ W − ( q + m − ω ) ¯ W + p ˆ W − p ¯ W , (103) C = − p ˆ W − p ¯ W + ( m − q − ω ) ˆ W + ( m + q − ω ) ¯ W , C = ( n − q − ω ) ˆ W + ( n + q − ω ) ¯ W − p ˆ W − p ¯ W , each one representing a combination of the modes S and R ; from (102) wehave ω = ω < ω = ω . (104)All together we have ω < ω , ω < ω < ω , (or ω > ω > ω ) , ω = ω < ω = ω , (105)and besides this we cannot give a complete ordering between these frequencieswithout the knowledge of the numerical values of components of B .23 .2.2. Classes mm , , ¯42 m and /mm For these classes the matrix B is reduced by the three subspaces: Z ≡ span { W , W , W } , Z ≡ span { W , W } , Z = U , , (106)with Lin ≡ Z ⊕ Z ⊕ Z .To find the solution in Z we notice that since ω = a − d is a root forthe cubic characteristic equation, then we can easily obtain: ω = a − d ,ω = a + b + d − (cid:114) ( a + d − b + 2 e , (107) ω = a + b + d (cid:114) ( a + d − b + 2 e , with ω < ω , ω < ω . (108)The corresponding (non-normalized) eigentensors are: C = W − W , C = α W + β W + γ W , (109) C = β W + α W + γ W , with: α = ( b − ω ) d − e , β = e − b ( a − ω ) , γ = e ( a − d − ω ) . (110)In the subspace Z we have two eigencouples associated a shear in theplane orthogonal to the c -axis ( C ) and a rotation about the propagationdirection ( C ): ( ω = c − f , C = ¯ W ) , (111)( ω = c + f , C = ˆ W ) , whereas the solutions on Z are given by (102) and (103) with p = 0; againwe have ω = ω < ω = ω . (112)and for B > ω < ω , (113)24he inequality being reversed when B < Z either D , or the traceless plane strain D ; in Z we have instead a shear inthe plane orthogonal to the c -axis and a rigid microrotation about the samedirection whereas in Z we have the same kinematics as in U .Also for these classes it is not possible to give a complete ordering betweenall the nine eigenvalues (108), (112) and (113), in absence of the numericalvalues for the components of B . c -axis Classes , ¯4 and /m . We begin with the lower-symmetry classes; inthis case the blocks of the matrix A have the following non-null components A ≡ • • · · · · • • · • • ·• • · · · · • • · • • ·· · • • • • · · • · · •· · • • • • · · • · · •· · • • • • · · • · · •· · • • • • · · • · · •• • · · · · • • · • • ·• • · · · · • • · • • ·· · • • • • · · • · · •• • · · · · • • · • • ·• • · · · · • • · • • ·· · • • • • · · • · · • , (114)with the independent components given by (158), (166), (162) and (173)evaluated for m = m = 0 and m = 1.The matrix (114) is reduced by the pairs M and M : M ≡ span { e } ⊕ U , M ≡ span { e , e } ⊕ U , (115)were U and U are defined by (93). 25e begin with the subspace M and define the normalized components: a ( ξ ) = ξ A ,b ( ξ ) = ξ P L c L m √ J + iξ Q √ J , c ( ξ ) = ξ P L c L m √ J + iξ Q √ J ,d ( ξ ) = ξ A L c J + B ρJ , e ( ξ ) = ξ A L c J + B ρJ ,f ( ξ ) = ξ A L c J + B ρJ , g ( ξ ) = ξ A L c J + B ρJ , (116) h ( ξ ) = ξ A L c √ J J + B ρ √ J J , l ( ξ ) = ξ A L c √ J J + B ρ √ J J ,n ( ξ ) = ξ A L c J + B ρJ , p ( ξ ) = ξ A L c J + B ρJ ,m ( ξ ) = ξ A L c √ J J + B ρ √ J J , q ( ξ ) = ξ P L c L m √ J + iξ Q √ J ;then, since the characteristic equation can be factorized into a second- anda fourth-grade algebraic equations, we obtain by the means of the Cardano’sformulae for the fourth-grade algebraic equations [29], the explicit represen-26ation of the six eigenvalues ω = 12 (cid:16) F − √ F − G (cid:17) ,ω = 12 (cid:16) F + √ F − G (cid:17) ,ω = 14 A − (cid:114) A + 23 B + W − (cid:118)(cid:117)(cid:117)(cid:116) A + 43 B − W − S (cid:113) A + B + W ,ω = 14 A − (cid:114) A + 23 B + W (117)+ 12 (cid:118)(cid:117)(cid:117)(cid:116) A + 43 B − W − S (cid:113) A + B + W ,ω = 14 A + 12 (cid:114) A + 23 B + W − (cid:118)(cid:117)(cid:117)(cid:116) A + 43 B − W + S (cid:113) A + B + W ,ω = 14 A + 12 (cid:114) A + 23 B + W + 12 (cid:118)(cid:117)(cid:117)(cid:116) A + 43 B − W + S (cid:113) A + B + W , A = a + d − e + f + g + h ,B = cc (cid:63) + 2( bb (cid:63) + l + m + qq (cid:63) ) + ( n + p ) − ( d − e )( a + f + g + h ) − ( f + h )( a + g ) − ag ,C = 2 bb (cid:63) ( d − e + g ) + 2 l ( a + d − e ) + 2 m ( a + f + h ) + 2 qq (cid:63) ( f + g + h )+ ( a + g )(( n + p ) − ( f + h )( d − e )) + ( cc (cid:63) − ag )( d − e + f + h ) (118) − lm + qb (cid:63) )( n + p ) + ( lb + mq ) c (cid:63) ) ,D = 2 bb (cid:63) ( m − g ( d − e )) + 2 qq (cid:63) (2 l − g ( f + h )) + 2 l (2 bc (cid:63) − al )( d − e )+ 2 m (2 cq (cid:63) − am )( f + h ) + (( n + p ) − ( f + h )( d − e ))( cc (cid:63) − ag )+ 4( n + p )( m ( cb (cid:63) + al ) + q ( gb (cid:63) − c (cid:63) l )) − lmqb (cid:63) F = d + e + f − h ,G = ( d + e )( f − h ) − ( n − p ) , and P = − B − ACB + 72 DB + 27 C + 27 A D ,Q = B + 3 AC + 12 D ,S = A + 4 AB − C ,W = (cid:113) P + (cid:112) P − Q √ √ Q (cid:113) P + (cid:112) P − Q . By looking at (117), by (116) and (118) we notice that the frequencies ω , depend solely on the components of A and B , whereas the others dependalso on the acoustic tensors A , P and Q .The associated eigenvectors: w = { a o = α e ; V = β W + γ W + δ W + ( θ + (cid:15) ) ˆ W + ( (cid:15) − θ ) ¯ W } ,w = { a o = α e ; V = γ W + β W + δ W − ( θ + (cid:15) ) ˆ W + ( (cid:15) − θ ) ¯ W } ,w = { a o = α e ; V = β ( I − W ) + δ W + 2 (cid:15) ¯ W } , (119) w = { a o = α e ; V = β W + γ W + δ W + ( β − γ ) ˆ W + ( β + γ ) ¯ W } ,w = { a o = α e ; V = γ W + β W + δ W − ( β − γ ) ˆ W + ( β + γ ) ¯ W } ,w = { a o = α e ; V = β ( I − W ) + δ W + 2 (cid:15) ¯ W } , D , S , R or with the modes D and R : the coefficients of (119) are given in § .5.5.2of the Appendix.For ξ → a , b , c and q vanish, then from (117) and the results of § .5.5.2 we have that: ( ω , ( ξ ) , w , ) → ( ω , , { , C , } ) , ( ω ( ξ ) , w ) → ( ω , { , C } ) , (120)( ω , ( ξ ) , w , ) → ( ω , , { , C , } ) , ( ω ( ξ ) , w ) → (0 , { e , } ) , with the cut-off frequencies ω , and ω , , given respectively by (98) and(99) and where the microdistortions C − are given by (101); therefore inthe subspace M there are one acoustic longitudinal wave and five opticwaves with cut-off frequencies (98) and (99).A qualitative graph of the dispersion relations ω = ω ( ξ ) for the solu-tions in the subspace M , which can be obtained for arbitrary values of thecomponents, is given in Fig. 3.3.1.Fig. 3.3.1. Schematic of the dispersion relations in M . The dotted linerepresent the linearly elastic longitudinal wave. AL=Acoustic longitudinalwave; OL , , , =optic longitudinal waves, modes D , S , R ; OL =optic lon-gitudinal wave, modes D , R . The frequencies on the ω -axis are the cut-offfrequencies (98), (99).
29n the subspace M , provided we define the normalized components: a ( ξ ) = ξ A , b ( ξ ) = ξ A ,c ( ξ ) = ξ P L c L m √ J + iξ Q √ J , d ( ξ ) = ξ P L c L m √ J + iξ Q √ J ,e ( ξ ) = ξ A L c J + B ρJ , f ( ξ ) = ξ A L c J + B ρJ ,g ( ξ ) = ξ A L c √ J J + B ρ √ J J , h ( ξ ) = ξ A L c √ J J + B ρ √ J J , (121) m ( ξ ) = ξ P L c L m √ J + iξ Q √ J , n ( ξ ) = ξ P L c L m √ J + iξ Q √ J , we notice that the characteristic equation can be factorized into two cubicequations in ω . Accordingly the eigenvalues can be obtained by using twicethe Cardano’s formula: ω = 13 ( A + 2 (cid:112) P cos θ ,ω = 13 ( A + 2 (cid:112) P cos θ − π ,ω = 13 ( A + 2 (cid:112) P cos θ + 2 π ,ω = 13 ( A + 2 (cid:112) P cos θ , (122) ω = 13 ( A + 2 (cid:112) P cos θ − π ,ω = 13 ( A + 2 (cid:112) P cos θ + 2 π , where A , = a ± b + e + f ,B , = 2 a ± a ( b ∓ e ∓ f )+ 3 a (2 b + 3( cc ∗ + dd ∗ ) − ( e − f ) − g + h ) + 3( mm ∗ + nn ∗ ) ∓ b ( e + f ) + 4 ef ) ± b ± e ± f ) (123) ± b ( cc ∗ + dd ∗ − e − f ) + 9( g + h )( ∓ b + e + f ) ± mm ∗ ( b ± e ∓ f ) ± nn ∗ ( b ± f ∓ e ) − e ( b + f − cc ∗ + 6 dd ∗ ) − f ( b + 6 cc ∗ − dd ∗ + e + 4 be ) + 54( g ( cd ∗ + mn ∗ ) + h ( dm ∗ − cn ∗ )) C , = ∓ cc ∗ − ( dd ∗ + g + h + mm ∗ + nn ∗ ) + ( a ± b )( e + f ) + ef , θ α = cos − B α √ P α , P α = A α − C α , α = 1 , . (124)In this case all the frequencies depend on all the components of the blockmatrix A ; as far as the corresponding eigenvectors are concerned, they are w k = { a ko = α k e + β k e ; V k = ( γ k + (cid:15) k ) ˆ W + ( δ k + θ k ) ˆ W (125)+ ( γ k − (cid:15) k ) ¯ W + ( δ k − θ k ) ¯ W } , k = 7 , . . . , the coefficients being given in detail into the Appendix, § .5.5.3. The kine-matics described by these eigenvectors is formed by a macroscopic transversewave coupled with a combination of the modes S and R .The behavior of the eigencouples (122), (125) for ξ →
0, since in such acase a = b = c = d = m = n = 0, yields four optic waves (with multiplicity2) with cut-off frequencies (102) and eigenvectors given by (103): w (0) = w (0) = { ; C , } , w (0) = w (0) = { ; C , } , (126)and two acoustic waves with eigenvectors w (0) = { e − e ; } , w (0) = { e + e ; } . (127)As we did for M we give in Fig.3.3.1 a representative graph for thedispersion relations in M . 31ig. 3.3.1. Schematic of the dispersion relations in M . The dotted linerepresent the linearly elastic transverse waves. AT , =Acoustic transversewave; OT , , , =optic transverse waves, modes S , R . The frequencies onthe ω -axis are the cut-off frequencies (102). To summarize, for the Tetragonal classes 4 , ¯4 , /m , for a propagation di-rection m along the tetragonal c -axis, we have three Acoustic and nine Opticwaves which depend on 43 independent components of A (three componentsof A , thirteen for both A and B , seven for both P and Q ):(AL) One Acoustic wave associated with a macroscopic displacement along c and a combination of the modes D , S and R , which for ξ = 0reduced to a macroscopic longitudinal wave;(AT , ) Two Acoustic waves associated with a macroscopic displacement or-thogonal to c , coupled a combination of the modes S and R . For ξ = 0 these waves reduce to two macroscopic orthogonal transversewaves;(OL , , , ) Four Optic waves associated with a macroscopic displacement along c and with a microdistortion which combines the modes D , S and R which for ξ = 0 reduces to the pure microdistortions (101) , , , ;(OL ) One Optic wave associated with a macroscopic displacement along c and with a combination of the modes D and R ;(OT , , , ) Four Optic waves associated with a macroscopic displacement orthog-onal to c coupled with a combination of the S and R modes, whichfor ξ = 0 reduce to the shear microdistortion (103) and to the rigidrotation (103) . Classes mm , , ¯42 m and /mm . When we deal with the high-symmetric tetragonal classes, for a propagation direction along the tetragonal c -axis with m = e , the matrix A has the following non-null components,the independent ones given by relations (159), (165), (170) and (174) of the32ppendix, evaluted for m = m = 0 and m = 1. A ≡ • • · · · · · • · · • ·• • · · · · • · · • · ·· · • • • • · · · · · ·· · • • • • · · · · · ·· · • • • • · · · · · ·· · • • • • · · · · · ·· • · · · · • · · • · ·• · · · · · · • · · • ·· · · · · · · · • · · •· • · · · · • · · • · ·• · · · · · · • · · • ·· · · · · · · · • · · • : (128)accordingly A is reduced by the three subspaces: N ≡ span { e } ⊕ Z , N ≡ Z , N ≡ span { e , e } ⊕ Z ; (129)We begin our analysis with the subspace N and notice that the eigen-values can be obtained from those in M when we set d = e = 0 and m = n = p = q = 0 into (116). Then the characteristic equation admits theroot ω = f − h and thus, by the means of the Cardano’s formulae ( vid. e,g. [29]) we obtain the four eigenvalues (for once we write one of them in termscomponents and characteristic length): ω ( ξ ) = ξ ( A J + A √ J J ) L c + B ρJ + B ρ √ J J ,ω ( ξ ) = 13 ( A + 2 √ P cos θ , (130) ω ( ξ ) = 13 ( A + 2 √ P cos θ + 2 π ,ω ( ξ ) = 13 ( A + 2 √ P cos θ − π , where P = A − B , Q = 2 A − AB − C , θ = cos − Q √ P , (131)and
A, B and C are obtained by setting d = e = m = n = p = q = 0 into(118). The corresponding (non-normalized) eigenvectors are w ( ξ ) = { ; V = W − W } , (132) w ( ξ ) = { α e ; V = β W + γ W + δ W } ,w ( ξ ) = { α e ; V = γ W + β W + δ W } ,w ( ξ ) = { α e ; V = β ( I − W ) + δ W } , α i , β j , γ k and δ k are given in § .5.5.4 of theAppendix.Looking at (130) and (132) we notice first of all that the frequencies ω , , ( ξ ) are associated with the mode D coupled with a macroscopic lon-gitudinal wave; the frequency ω ( ξ ) is instead associated uniquely to thetraceless real microdistortion D .In the limit ξ →
0, since a , b and c vanish, then the three frequencies ω ( ξ ) and ω , ( ξ ) reduce to (107) with eigenvectors w (0) = { ; C } , w (0) = { ; C } , w = { ; C } , (133)with the three microdistortions given by (109): these are optic frequencies,the values (107) being the associated cut-off values; the frequency ω ( ξ ) van-ishes instead for ξ → w (0) = { e ; } , (134)which represents a purely macroscopic acoustic longitudinal wave.The solutions in the subspace N are are the same as those in Z withfrequencies ω ( ξ ) = ξ L c A − A J + B − B ρJ , (135) ω ( ξ ) = ξ L c A + A J + B + B ρJ , and accordingly describe optic waves with purely microdistortion amplitudesand whose cut-off frequencies are given by (111); the corresponding realeigenvectors are: w (0) = { ; V = C } , w (0) = { ; V = C } , (136)with the microdistortions given by (111).We finish with the subspace N where the solutions are obtained by set-34ing m = n = h = 0 into (121) which yield the six eigenvalues ω = 13 ( A + 2 (cid:112) P cos θ ,ω = 13 ( A + 2 (cid:112) P cos θ + 2 π ,ω = 13 ( A + 2 (cid:112) P cos θ − π ,ω = 13 ( A + 2 (cid:112) P cos θ , (137) ω = 13 ( A + 2 (cid:112) P cos θ + 2 π ,ω = 13 ( A + 2 (cid:112) P cos θ − π , where A α and P α for α = 1 , m = n = h = 0into (123). The eigenvector have the same representation (125) with thecomponent once again obtained by putting m = n = h = 0 into those givenin § .5.5.3: w = { a o = α e + β e ; V = ( γ + (cid:15) ) ˆ W + ( δ + θ ) ˆ W + ( γ − (cid:15) ) ¯ W + ( δ − θ ) ¯ W } ,w = { a o = α e + β e ; V = ( γ + (cid:15) ) ˆ W + ( δ + θ ) ˆ W + ( γ − (cid:15) ) ¯ W + ( δ − θ ) ¯ W } ,w = { a o = β e + α e ; V = ( δ + θ ) ˆ W + ( γ + (cid:15) ) ˆ W − ( δ − θ ) ¯ W − ( γ − (cid:15) ) ¯ W } ,w = { a o = β e + α e ; (138) V = ( δ + θ ) ˆ W + ( γ + (cid:15) ) ˆ W − ( δ − θ ) ¯ W − ( γ − (cid:15) ) ¯ W } ,w = { a o = α e + β e ; V = ( γ + (cid:15) ) ˆ W + ( δ + θ ) ˆ W + ( γ − (cid:15) ) ¯ W + ( δ − θ ) ¯ W } ,w = { a o = β e + α e ; V = ( δ + θ ) ˆ W + ( γ + (cid:15) ) ˆ W − ( δ − θ ) ¯ W − ( γ − (cid:15) ) ¯ W } , the coefficients being given in detail into the Appendix, § .5.5.5. These eigen-vectors describe a kinematics formed by a macroscopic transverse wave cou-pled a combination of the modes S and R .35he eigenvalues (137) and their associated eigenvectors becomes, sincefor ξ → a = b = c = d = 0: w , , , (0) = { ; C , , , } , (139)which correspond to four optic waves (with multiplicity 2) with cut-off fre-quencies (102) with B = 0 and eigenvectors given by (103) and two acousticwaves with eigenvectors: w (0) = { e − e ; } , w (0) = { e + e ; } . (140)To summarize, in the high-symmetric tetragonal classes 4 mm , , ¯42 m and 4 /mm , for the propagation direction along the tetragonal c -axis we havethe three Acoustic and nine Optic waves, which depend on the 25 independentcomponents of A (three of A , seven each of A and B and four each of P and Q ):(AL) One Acoustic wave associated with a macroscopic displacement along c and the mode D which for ξ = 0 yields a macroscopic longitudinalwave;(AT , ) Two Acoustic waves associated with a macroscopic displacemente or-thogonal to c , coupled with the S and D modes. For ξ = 0 theyreduce to two macroscopic orthogonal transverse waves;(OL , , ) Two Optic waves associated to a macrodisplacement along e coupledwith the mode D , which for ξ = 0 reduces to the pure microdistortions(109) , ;(OD) One Optic wave associated to the traceless distortion D which is inde-pendent on the macroscopic displacement and which reduces, for ξ → ;(OS) One Optic wave associated to the the mode S which is independenton the macroscopic displacement and reduces for ξ → ;(OR) One Optic wave associated to the mode R , which is independent onthe macroscopic displacement and reduces for ξ → ;(OT , , , ) Four Optic waves associated with a macroscopic displacement orthog-onal to c coupled with a combination of the modes S and R , whichfor ξ = 0 reduce to the shear microdistortions (103) and to the rigidrotations (103) with B = 0. 36s far as the eigenvalues ordering is concerned, we can only say that: ω < ω ,ω < ω < ω , ( or ω > ω > ω ) , (141) ω < ω < ω , ( or ω > ω > ω ) ,ω < ω < ω , ( or ω > ω > ω ) . c -axis When the propagation direction is orthogonal to the tetragonal c -axis, wemay assume into the constitutive relations (158), (159), (162), (164), (166),(170), (173) and (174) that m = cos θ , m = sin θ and m = 0. An inspectionof these relations however shows that the tensor A ( θ ) has six independentcomponents, the tensor A ( θ ), twenty-six independent components and thatthe tensor ξ P + iξ Q has twenty-seven independent components for a genericvalue of θ . It is indeed this last block of A which makes the problem afully-coupled one: further, for all classes but two, there is no great differencebetween the classes and the propagation condition is coupled as in the caseof any cristal of the Triclinic group. Hence, the best we can say is that therewill be three acoustic and nine optic waves with cut-off frequencies givenby the solution of the microvibrations problem of § .3.2 and that, despitethe fact that we can still write the explicit expressions for the eigencouples,these relations will be nearly useless given the total kinematical coupling.Therefore we do not find useful to pursue the matter any further.However, for the two centrosymmetric classes 4 /m and 4 /mmm the tensor P vanishes altogether since it depends on the components of an odd-tensorand for the class 4 /mm the matrix A has only to forty-one independentcomponents: M ≡ • • · • • • · · • · · •• • · • • • · · • · · •· · • · · · • • · • • ·• • · • • • · · • · · •• • · • • • · · • · · •• • · • • • · · • · · •· · • · · · • • · • · ·· · • · · · • • · · • ·• • · • • • · · • · · •· · • · · · • · · • • ·· · • · · · · • · • • ·• • · • • • · · • · · • , (142)and more important it is reduced by the pairs L ≡ span { e } ⊕ U , L ≡ { e , e } ⊕ U . (143)37n the subspace L the fifth-order characteristc equation in ω dependson 15 independent components (which may reduce to 13 when either θ = 0or θ = π/ e and a combination of the modes S and R . In the limit ξ → L we have seven eigenvectors representing amacroscopic displacement orthogonal to e coupled with a combination of the D k , S and R modes, which in the limit ξ → c -axis the role of the subspaces U and U associated withthe microvibrations is exchanged with respect of the case m = e .
4. Conclusions
For a bulk three-dimensional crystalline body we modeled the interac-tion between macroscopic and lattice waves by means of a continuum withaffine-structure, or micromorphic. The propagation condition we obtained isdescribed by a twelve-dimensional hermitian acoustic matrix A , the eigen-vector being elements of V ⊕
Lin which represent three macroscopic dis-placements and nine lattice microdistortions. We showed that for such apropagation condition admits three acoustic and nine optic waves for anycrystalline symmetry; moreover it depends on three length scale represent-ing the crystal lattice inhomogeneity, the size of the bulk crystal and theeffect of lattice inertia. In terms of these length and their ratios we obtainedtwo limit problems which recover the long wavelength approximation and the microvibration problems, well-known and studied into details for isotropicmaterials.Then we investigated in full detail the propagation condition for crystalsof the Tetragonal point group: the the matrix A is reduced by two or threesubspaces of V ⊕
Lin, depending on the classes, whereas in the microvibra-tions limit case the propagation condition is determined by B which is inturn reduced by two or three subspaces of Lin. The relations between thesesubspaces and the Tetragonal classes is represented schematically in Fig.4.38 (cid:11)(cid:10) (cid:8)(cid:9) U (cid:11)(cid:10) (cid:8)(cid:9) M M N N N Z (cid:11)(cid:10) (cid:8)(cid:9) Z (cid:11)(cid:10) (cid:8)(cid:9) Z (cid:11)(cid:10) (cid:8)(cid:9) −→−→−→←−←− , ¯4 , /m mm, , ¯42 m, /mm Fig. 4.
The subspaces M j ⊂ V ⊕ Lin and N k ⊂ V ⊕ Lin reduced by A andtheir relation with the subspaces U j ⊂ Lin and Z k ⊂ Lin reduced by B . The results we obtained for the two limit problems are represented inTable 1: it is remarkable that in the long-wavelength approximation themacroscopic displacements are real whereas the lattice microdistortions arepurely immaginary; conversely, in the microvibration problem the matrix A reduces to a real one and the lattice microdistortions are real. In both caseswe find that the lattice modes can be described by the means of three kindof dilatations, two shear deformations and two rigid rotations.When we deal with the complete propagation condition in the case ofwaves propagating along the tetragonal c -axis, in all cases but three themodes are complex and fully-couples macroscopic displacements with latticemicrodistortions: only for the highly-symmetric classes we have three fullyoptic modes with real microdistortions without macroscopic displacements.These results are represented in Table 2:We finish by giving an insight for waves propagating in directions orthog-onal to the c -axis which, in a remarkeble difference with the previous caseyields a fully-coupled problem as in the fully anisotropic case.As far as we know, this is the most complete analysis for the wave propa-gation problem in classical Tetragonal micromorphic continua: of course, tomake such an analyisis a predictive tool for experimental applications we needto know a complete set of parameters (43 for the low-symmetric classes 4, ¯4and 4 /m and 25 for the high-symmetry classes ). These parameters whichare very difficult to obtain and evaluate by the means of a set of simple ex-periment can be obtained, at the present only by two viable approach, that39 able 1: Modes in the Long-wavelength (LW) and Microvibration (M) approximations. m e e e D D D S S R R · · (cid:60) L · (cid:61) · · · (cid:61)† · LW (cid:107) e (cid:60) T · · · · · · (cid:61) · (cid:61)· (cid:60) T · · · · · (cid:61) · (cid:61)· · (cid:60) T · · · · (cid:61) · (cid:61) LW ⊥ e (cid:60) · · [ (cid:61) ] ( (cid:61) ) · (cid:61) · (cid:61) ·· (cid:60) · [ (cid:61) ] ( (cid:61) ) · (cid:61) · (cid:61) · (2) · · · (cid:60) · · (cid:60)◦ · (cid:60)• · (2) · · · (cid:60) · · (cid:60)◦ · (cid:60)• · M · · · · (cid:60) · · · (cid:60) · LS (2) , (cid:91) · · · · · · · (cid:60) · (cid:60) (2) , (cid:91) · · · · · · · (cid:60) · (cid:60)· · · · · (cid:60) · · · · (2) · · · (cid:60) · · · · · · M · · · · · · (cid:60) · · · HS · · · · · · · · (cid:60) · (2) , (cid:91) · · · · · · · (cid:60) · (cid:60) (2) , (cid:91) · · · · · · · (cid:60) · (cid:60) HS=classes 4 mmm , 422, 4 /mm , ¯42 m , LS=classes 4, ¯4, 4 /m ; (cid:60) =Real component, (cid:61) =Immaginary com-ponent; L =longitudinal wave, T =transverse wave; • =equal values components; ◦ =equal value, oppositesign components; ( (cid:93) )=number of independent eigentensors with same structure; (cid:91) =multiple eigenvalues; † vanishes for HS classes; [ (cid:61) ] = only LS classes; ( (cid:61) ) = only HS classes. able 2: Modes, wave propagation along the direction c . e e e D D D S S R R ξ → · · • C L C · · C ◦ · C • · OL , (2) · · • C L C · · C ◦ · C • · OL , LS · · C L · C · · · C · OL · · C L · C · · · C · AL(4) C T C T · · · · · C · C OT − (2) C T C T · · · · · C · C AT − · · · · · (cid:60) · · · · OL · · • C L C · · · · · · OL · · • C L C · · · · · · OL · · C L · C · · · · · ALHS · · · · · · (cid:60) · · · OS · · · · · · · · (cid:60) · OR(2) C T C T · · · · · C · C OT , (2) C T C T · · · · · C · C OT , (2) C T C T · · · · · C · C AT , mmm , 422, 4 /mm , ¯42 m , LS=classes 4, ¯4, 4 /m ; C =Complex components; L =longitudinalwave, T =transverse wave; • =equal values components; ◦ =equal value, opposite sign components;( (cid:93) )=number of independent eigentensors with same structure.
41s the numerical homogeneization approach used into [28] for relaxed micro-morphic tetragonal continua undergoing plane strain, or by homogeneizationand identification methods from lattice dymanics, as it was done for cubicDiamond crystals and Silicon into [17]. Such a technique shall be appliedin a future paper to evaluate the components of the acoustic matrix A forthe Tetragonal 4 /m crystals of PbWO (PWO), widely used in high-energyphysics as ionizing radiation detectors. Acknowledgments
The research leading to these results is within the scope of CERN R&DExperiment 18 ”Crystal Clear Collaboration” and the PANDA Collaborationat GSI-Darmstadt. The author declares that he would never have dared touse repeatedly Cardano’s formulae for the third- and fourth-order algebraicequations, if he hadn’t been locked down by the COVID-19.
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On the Construction of Matter Tensors in Crystals , ActaCrystallographica, , 15–20, (1953).[33] M. Olive, N. Auffray, Symmetry classes for odd-order tensors . Journalof Applied Mathematics and Mechanics / Zeitschrift f¨ur AngewandteMathematik und Mechanik, (5), 421–447, (2014).[34] M. Olive, N. Auffray, Symmetry classes for even-order tensors , Mathe-matics and Mechanics of Complex Systems, vol. (2), 177–210, (2013).[35] N. Auffray, Q.-C. He, H. Le Quang,. Complete symmetry classificationand compact matrix representations for 3D strain gradient elasticity . Int.J. of Solids and Structures, , 197–210, (2019).[36] H. Le Quang, Q.-C. He,
The number and types of all possible rotationalsymmetries for flexoelectric tensors , Proc. Royal Society A, , 2369–2386, (2011). 45 . Appendix
In the first part of this Appendix ( §§ .5.1-5.3) we shall list, in tabular form when it is possible, the independentcomponents of the fourth-order tensors C , D and B , of the fifth-order tensors F and G and of the sixth-order tensor H which appear into the constitutive relations (36), for all the classes of the Tetragonal point group. For the fourth-ordertensors we refer to [30] whereas for the other tensors we refer to the results obtained into [31] and [32] (The most recentresults given e.g. into [33], [34], [35] and [36] cannot be applied to the present case since they were obtained for tensorsendowed with some minor symmetries which in our case are missing). We shall list also, for convenience, the independentcomponents of the microinertia fourth-order tensor J .In the second part of this Appendix, § .5.4, we shall list, for the Tetragonal point group, the independent componentsof the acoustic tensors A ( m ), ˆ A ( m ), P ( m ), Q ( m ) and A ( m ) showing, by the means of (55), the explicit dependence ofthese components on the propagation direction m and on the tensorial quantities listed in the first part.In the third part, § .5.5, we shall list the coefficients of the linear combinations between the elements of the bases { e k } and { W h } , k = 1 , , h = 1 , . . . , The non-zero components for all the classes of the Tetragonal point group are given in tabular form into [30] for thetensors C and B whereas those of D can be obtained with the additional conditions induced by the symmetries of the firsttwo components.We list the tabular form of these tensors in the Voigt’s notation 1 = 11 , , , , , , C All classes (6 independent components):[ C ] ≡ C C C · C C · · C · · · C · · · · C · · · · · C . (144)The tensor C micro has the same non-null components as (144). D Classes 4, ¯4 and 4 /m (14 independent components):[ D ] ≡ D D D D − D D D D D − D D D D D − D D D D − D
00 0 0 D D − D D D − D D D D D D − D
00 0 0 D D − D D D − D D D ; (145)Classes 4 mmm , 422, 4 /mm and ¯42 m (8 independent components):[ D ] ≡ D D D D D D D D D D D D D
00 0 0 0 0 D D D D D D
00 0 0 0 0 D D ; (146) .1.3. The tensor B Classes 4, ¯4 and 4 /m (13 independent components):[ B ] ≡ B B B B − B · B B B − B · · B B − B · · · B B B · · · · B B · · · · · B B · · · · · · B − B · · · · · · · B · · · · · · · · B ; (147)Classes 4 mmm , 422, 4 /mm and ¯42 m (9 independent components):[ B ] ≡ B B B · B B · · B · · · B B · · · · B B · · · · · B B · · · · · · B · · · · · · · B · · · · · · · · B ; (148) J Let J be the micro-inertia tensor whose components are J ij = J ji , i, j = 1 , ,
3: then, by (47) , the matrix J = J T is [ J ] ≡ J J J · J J J · · J J J · · · J J · · · · J J · · · · · J J · · · · · · J · · · · · · · J · · · · · · · · J ; (149)for Tetragonal crystals, provided we identify the c − axis with the direction e , we have J = J and J ij = 0 , i (cid:54) = j :accordingly (149) reduces to:[ J ] ≡ J · J · · J · · · J · · · · J · · · · · J · · · · · · J · · · · · · · J · · · · · · · · J . (150) A detailed study of the symmetries for fifth- and sixth-order tensor was done into [31]: for the Tetragonal group thesymmetries are different between classes, and accordingly we study them in detail beginning with the fifth-order tensor G ; we also follow [31] into the use of the notation G [5] , (151)to denote all the 5 possible combinations of the index, namely: G , G , G , G and G .Since G is an odd tensors, then for the two centrosymmetric classes 4 /m and 4 mm their components vanishesaltogether: for the other classes we have the following restrictions. • Class 4: for this class we have 61 independent components: G [1] , G = G [5] , G = G [10] , (152) G = −G [20] , G = −G [10] , G = G [15] . • Class ¯4: for this class we have the following restrictions into (152) G = 0 , G = G = 0 , (153)which means that (152) , must be zero and the independent components reduce to 50. Classes 4 /mm , ¯42 m , 422: for these classes the tensors G splits into polar and axial ones.The polar tensors have ha 61 non-zero components: G [1] G = G [5] , (154) G = G [10] , G = G [15] ,
31 being the independent ones.The axial tensors have 60 components with only 30 independent for the class 4 /mm : G = −G [20] , G = −G [10] ; (155)the components for the class ¯42 m are obtained by changing the sign of the components (155), whereas those forthe classes 422 are obtained by setting to zero (155).To obtain the number of the independent components for the tensor F , we recall that it obeys F ijhkm = F jihkm and hence the number of independent components reduces to 41 components for the class 4, 33 components for class ¯4and 42 components (21 independent) for the classes 4 /mm , ¯42 m and 422. H Also for sixth-order tensors the symmetries changes for different classes. Following [31] we have, by using the sameconvention we used for the fifth-order tensor, that the for the classes 4, ¯4 and 4 /m the non-null components are: H = H [1] H [1] H = − H [6] , H = H [15] , H = H [15] , H = H [15] , H − H [15] , (156) H [20] , H = − H [10] , H = H [60] , H = H [45] ;however, the number of non-zero and independent component obtained into [31] refers to a tensor with no major symme-tries, whereas in our case H = H T and hence there are only 108 independent components into (156).For the classes 422, 4 mm , 4 /mmm and for the polar tensors of the class ¯42 m we have the further restrictions into(156): H = H = H = H = 0 , (157) H = H = H = H = 0 , which further reduce the number of independent components. A ( m ) From relation (55) we have the components of the second-order acoustic tensor A ( m ) for the classes 4 , ¯4 and 4 /m of the Tetragonal group: A = ρ − (cid:16) ( C + B + 2 D ) m + ( C + B + 2 D ) m + ( C + B + D + D ) m + ( C + B − B + D + 2 D − D ) m m (cid:17) A = ρ − (cid:16) ( C + B + 2 D ) m + ( C + B + 2 D ) m + ( C + B + D + D ) m − ( C + B − B + D + 2 D − D ) m m (cid:17) (158) A = ρ − (cid:16) ( C + D + D + B )( m + m ) + ( C + 2 D + B ) m (cid:17) A = ρ − (cid:16) ( − B + D − D ) m m + ( C + B + B + 2 D + D + D ) m m (cid:17) A = ρ − (cid:16) ( C + B + B + 2 D + D + D ) m m + ( B + D − D ) m m (cid:17) A = ρ − (cid:16) ( C + B + D − D ) m + ( − C + B − D ) m + ( − B + D − D ) m + ( C + 2 C + B + B + 2( D + D )) m m (cid:17) . or the classes 4 mmm , , /mm and ¯42 m relations (158) simplify into: A = ρ − (cid:16) ( C + B + 2 D ) m + ( C + B + 2 D ) m + ( C + B + D + D ) m (cid:17) A = ρ − (cid:16) ( C + B + 2 D ) m + ( C + B + 2 D ) m + ( C + B + D + D ) m (cid:17) A = ρ − (cid:16) ( C + D + D + B )( m + m ) + ( C + 2 D + B ) m (cid:17) (159) A = ρ − (cid:16) ( C + 2 D + D + D ) m m + ( B + B ) m m (cid:17) ,A = ρ − (cid:16) ( C + 2 D + D + D ) m m + ( B + B ) m m (cid:17) ,A = ρ − (cid:16) C ( m − m ) + ( C + 2 C + 2 D + 2 D + B + B ) m m (cid:17) . ˆ A ( m ) The definition of the acoustic tensor ˆ A ( m ) leads to the same result as in the classical linearly elastic case with thecomponents of ˆ C = C − C micro in place of those of C , accordingly, since A ij = ρ − C iljk m k m l , then for the classes 4, ¯4and 4 /m we have the explicit representation:ˆ A = ρ − (ˆ C m + ˆ C m + ˆ C m m + ˆ C m ) , ˆ A = ρ − (ˆ C m + ˆ C m − ˆ C m m + ˆ C m ) , ˆ A = ρ − ( m ˆ C + ˆ C ( m + m )) , (160)ˆ A = ρ − ˆ C m m , ˆ A = ρ − ˆ C m m , ˆ A = ρ − (ˆ C ( m − m ) + (ˆ C + 2ˆ C ) m m ) ;the components for the classes 4 mmm , 422, 4 /mm and ¯42 m can be obtained by taking ˆ C = 0 into (160) . Q ( m ) For the classes 4 , ¯4 and 4 /m the components of the third-order tensor Q ( m ), calculated by the means of (55) , obeythe restrictions Q = Q , Q = Q , Q = Q , Q = − Q , Q = − Q , Q = − Q ; (161)the 21 independent components are given in explicit by: Q = ρ − (( D + B ) m + ( D + B ) m ) , Q = ρ − B m , Q = ρ − (( D + B ) m − ( D − B ) m ) , Q = ρ − ( D + B ) m , Q = ρ − (( D + B ) m − ( D + B ) m ) , Q = ρ − D m , Q = ρ − (( D + B ) m − ( D + B ) m ) , Q = ρ − ( D + B ) m , Q = ρ − (( D + B ) m + ( D + B ) m ) , Q = ρ − ( D + B ) m , Q = ρ − (( D − B ) m + ( D + B ) m ) , Q = ρ − ( D + B ) m , Q = ρ − ( − ( D + B ) m + ( D + B ) m ) , Q = ρ − ( D + B ) m , Q = ρ − (( D + B ) m + ( D + B ) m ) , Q = ρ − (( D + B ) m − ( D + B ) m ) , Q = ρ − (( D + B ) m + ( D + B ) m ) , Q = ρ − ( D m + ( D + B ) m ) , Q = ρ − (( D + B ) m + ( D + B ) m ) , Q = ρ − ( − ( D + B ) m + ( D + B ) m ) , Q = ρ − (( D + B ) m − D m ) . (162)The tabular representation is[ Q ] = ( Q ) ( Q ) ( Q ) Q Q ( Q ) Q Q ( Q )( Q ) ( Q ) ( Q ) Q − Q ( Q ) Q − Q ( Q ) Q Q Q ( Q ) ( Q ) Q ( Q ( Q ) − Q (163) here the 14 components which depend solely m , m are represented within brackets ( · ), the other 13 componentsdepending only on m ..For the tetragonal classes 4 mmm , 422, 4 /mm and ¯42 m the 6 components Q , Q , Q , Q , Q and Q vanishes and we have the three conditions Q = Q , Q = Q and Q = Q . There are 18 independentcomponents: Q = ρ − ( D + B ) m Q = ρ − ( D + B ) m Q = ρ − ( D + B ) m Q = ρ − ( D + B ) m Q = ρ − ( D + B ) m Q = ρ − ( D + B ) m Q = ρ − ( D + B ) m Q = ρ − ( D + B ) m Q = ρ − ( D + B ) m Q = ρ − ( D + B ) m Q = ρ − ( D + B ) m Q = ρ − ( D + B ) m Q = ρ − ( D + B ) m Q = ρ − ( D + B ) m Q = ρ − ( D + B ) m Q = ρ − ( D + B ) m Q = ρ − ( D + B ) m Q = ρ − ( D + B ) m (164)with the tabular representation, where we put in square brackets [ · ] those depending on m and in round brackets ( · )those which depends only on m , the remaining depending on m :[ Q ] = [ Q ] [ Q ] [ Q ] 0 Q ( Q ) 0 Q ( Q )( Q ) ( Q ) ( Q ) Q Q ] Q Q ] Q Q Q [ Q ] ( Q ) 0 ( Q ) ( Q ) 0 . (165) P ( m ) We recall that for the centrosymmetric classes 4 /m and 4 mm since F = G = then P = . The components for theclass 4, evaluated by the means of (55) and (152) are (here we denote A = F + G ): P = ρ − (( A + A ) m m + ( A + A ) m m ) , P = ρ − (( A + A ) m m − ( A + A ) m m ) , P = ρ − (( A + A ) m m + ( A + A ) m m ) , P = ρ − ( A m − A m + A m + ( A + A ) m m ) , P = ρ − ( A m + A m + A m + ( A + A ) m m ) , P = ρ − (( A + A ) m m + ( A + A ) m m ) , P = ρ − ( A m − A m + A m + ( A + A ) m m + ( A + A ) m m ) , P = ρ − ( A m + A m + A m + ( A + A ) m m ) , P = ρ − (( A + A ) m m + ( A + A ) m m ) , P = ρ − (( A + A ) m m + ( A + A ) m m ) , P = ρ − ( − ( A + A ) m m + ( A + A ) m m ) , P = ρ − (( A + A ) m m − ( A + A ) m m + ( A + A ) m m ) , (166) P = ρ − ( A m + A m + A m + ( A + A ) m m ) , P = ρ − ( A m − A m − A m + ( A + A ) m m + ( A + A ) m m ) , P = ρ − (( A + A ) m m − ( A + ( A ) m m ) , P = ρ − ( A m + A m + A m − ( A + A ) m m ) , P = ρ − ( A m − A m − A m + ( A + A ) m m + ( A + A ) m m ) , P = ρ − (( A + A ) m m − ( A + A ) m m ) , P = ρ − ( A m + A m + A m + ( A + A ) m m + ( A + A ) m m ) , P = ρ − ( A m + A m + A m − ( A + A ) m m ) , P = ρ − ( A m + A m + A m + ( A + A ) m m ) , P = ρ − (( A + A ) m m + ( A + A ) m m ) , P = ρ − (( A + A ) m m − ( A + A ) m m ) , = ρ − ( A m − A m + A m + ( A + A ) m m − ( A + A ) m m ) , P = ρ − (( A + A ) m m + ( A + A ) m m ) , P = ρ − (( A + A ) m m − ( A + A ) m m ) , P = ρ − ( A m + A m − A m + ( A + A ) m m ) . In the tabular form of the tensor P ( m ) we show the components which are different from zero when either m = e or m · e = 0: [ P ] = P P P P
00 0 0 P P P P P P P P P . (167)In the case m = e we also have that for the class 4 P = P , P = − P , P = − P , P = − P , P = P , P = − P , (168)whereas for the class ¯4 the following components vanish: P = P = P = P = P = P = P = 0 . (169)For the remaining classes ¯42 m , 422, 4 /mm of the Tetragonal group we have that: P = ρ − ( A + A ) m m , P = ρ − ( A + A ) m m , P = ρ − ( A + A ) m m , P = ρ − ( A + A ) m m , P = ρ − ( A m + A m + A m ) , P = ρ − ( A + A ) m m , P = ρ − (( A + A ) m m + ( A + A ) m m ) , P = ρ − ( A m + A m + A m ) , P = ρ − ( A + A ) m m , P = ρ − ( A + A ) m m , P = ρ − ( A + A ) m m , P = ρ − (( A + A ) m m + ( A + A ) m m ) , (170) P = ρ − ( A m + A m + A m + ( A + A ) m m ) , P = ρ − (( A + A ) m m + ( A + A ) m m ) , P = ρ − ( A + A ) m m , P = ρ − ( A m + A m + A m ) , P = ρ − (( A + A ) m m + ( A + A ) m m ) , P = ρ − ( A + A ) m m , P = ρ − ( A m + A m + A m + ( A + A ) m m ) , P = ρ − ( A m + A m + A m ) , P = ρ − ( A m + A m + A m ) , P = ρ − ( A + A ) m m , P = ρ − ( A + A ) m m , P = ρ − (( A + A ) m m − ( A + A ) m m ) , P = ρ − ( A + A ) m m , P = ρ − ( A + A ) m m , P = ρ − ( A m + ( A + A ) m m ) . e show in the tabular form the components which are different from zero when m = e [ P ] = P P
00 0 0 P P P P P ; (171)whereas when m · e = 0 we have[ P ] = P P P P
00 0 P P P P P P P P P P . (172) A ( m ) For the classes 4, ¯4 and 4 /m by (55) and (156) we have: A = ρ − ( H m + H m + H m + 2 H m m ) , A = ρ − ( H m + H m + H m − H m m ) , A = ρ − ( H ( m + m ) + H m + 2 H m m ) , A = ρ − ( H m + H m + H m − H m m ) , A = ρ − ( H m + H m + H m + 2 H m m ) , A = ρ − ( H m + H m + H m + 2 H m m ) , A = ρ − ( H m + H m + H m − H m m ) , A = ρ − ( H m + H m + H m + 2 H m m ) , A = ρ − ( H m + H m + H m + 2 H m m ) , A = ρ − ( H m + H m + H m + 2 H m m ) , A = ρ − ( H m + H m + H m + 2 H m m ) , A = ρ − (2 H m m + 2 H m m ) , A = ρ − (2 H m m + 2 H m m ) , A = ρ − ( H m + H m + H m + 2 H m m ) , A = ρ − (2 H m m + 2 H m m ) , A = ρ − (2 H m m + 2 H m m ) , A = ρ − ( H m + H m + H m + 2 H m m ) , A = ρ − ( H m + H m + H m − H m m ) , A = − ρ − (2 H m m + 2 H m m ) , A = ρ − (2 H m m − H m m ) , A = ρ − ( H m − H m − H m + 2 H m m ) , A = − ρ − (2 H m m + 2 H m m ) , A = ρ − (2 H m m − H m m ) , (173) A = ρ − ( H m − H m − H m + 2 H m m ) , A = ρ − (2 H m m + 2 H m m ) , A = ρ − (2 H m m − H m m ) , A = ρ − ( H m − H m + H m + 2 H m m ) , A = ρ − (2 H m m + 2 H m m ) , A = ρ − (2 H m m − H m m ) , A = ρ − ( H m − H m − H m + 2 H m m ) , A = ρ − ( H m − H m − H m + 2 H m m ) , A = ρ − (2 H m m − H m m ) , A = ρ − ( H m + H m + H m − H m m ) , = ρ − ( H m − H m − H m m ) , A = ρ − (2 H m m − H m m ) , A = ρ − ( H m + H m + H m + 2 H m m ) , A = ρ − ( H m − H m + 2 H m m ) , A = ρ − ( H m + H m + H m + 2 H m m ) , A = ρ − (2 H m m + 2 H m m ) , A = ρ − (2 H m m + 2 H m m ) , A = ρ − (2 H m m + 2 H m m ) , A = ρ − ( H m + H m + H m + 2 H m m ) , A = ρ − ( H m + H m + H m + 2 H m m ) , A = ρ − (2 H m m + 2 H m m ) , A = ρ − (2 H m m + 2 H m m ) . From (173) we have that when m = e A has the same 13 independent components of of B . For m · e = 0 we haveinstead that there are 26 independent components.For the classes 422, 4 mm , 4 /mmm and for the polar tensors of the class ¯42 m , by using (157) into (173) we get A = ρ − ( H m + H m + H m ) , A = ρ − H m m , A = ρ − ( H m + H m + H m ) , A = ρ − H m m , A = ρ − ( H ( m + m ) + H m ) , A = ρ − H m m , A = ρ − ( H m + H m + H m ) , A = ρ − H m m , A = ρ − ( H m + H m + H m ) , A = ρ − H m m , A = ρ − ( H m + H m + H m ) , A = ρ − H m m , A = ρ − ( H m + H m + H m ) , A = − ρ − H m m , A = ρ − ( H m + H m + H m ) , A = ρ − H m m , A = ρ − ( H m + H m + H m ) , A = ρ − H m m , (174) A = ρ − ( H m + H m + H m ) , A = − ρ − H m m , A = ρ − ( H m + H m + H m ) , A = ρ − H m m , A = ρ − ( H m + H m + H m ) , A = ρ − H m m , A = ρ − ( H m + H m + H m ) , A = ρ − H m m , A = ρ − ( H m + H m + H m ) , A = ρ − H m m , A = ρ − ( H m + H m + H m ) , A = ρ − H m m , A = ρ − ( H m + H m + H m ) , A = ρ − H m m , A = ρ − ( H m + H m + H m ) , A = ρ − H m m , A = ρ − H m m , A = ρ − H m m , A = ρ − H m m , A = − ρ − H m m , A = ρ − H m m , A = ρ − H m m , A = ρ − H m m , A = ρ − H m m , A = ρ − H m m , A = ρ − H m m , A = ρ − H m m . For the direction of propagation m = e the tensor A has the same non-null independent components of B , whereas inthe case m · e = 0 there are 26 independent components. C k and the eigenvectors w k These components are calculated by the means of the free web application WIMS (wims.unice.fr). .5.1. Subspace U Here ¯ a = a − ω , ¯ b = b − ω and ¯ c = c − ω , the real coefficients a, b, c, d, e and f being defined by (95). All thecomponents must be divided by the norm (cid:107) C k (cid:107) = ( α k + β k + γ k + δ k + (cid:15) k ) / , k = 1 , . . . • C and C : α = − β = ( hd − ¯ ag )( l + ¯ bf ) + ( h ¯ a − gd )( l − ¯ b ¯ c ) + hg (2 el − ¯ bg ) + e ( f − ¯ c )( g + h ) + el ( g − h + ( f + ¯ c )(¯ a − d )) + ¯ bh ,β = − α = − ( h ¯ a + gd )( l + ¯ bf ) − ( hd + ¯ ag )( l − ¯ b ¯ c ) + e ( g + h )( f − ¯ c ) + ¯ bg ( h − g ) + el (2 gh − h + 3 g + ( f + ¯ c )( d − ¯ a )) ,γ = − γ = − l (¯ a + d )( h + g ) − egh ( h + g ) + e ( d − ¯ a )( f − ¯ c )( g − h ) + e ( h + g ) + l ( f + ¯ c )(¯ a − d ) ,δ = δ = (¯ a − d )(2 l + ¯ b ( f − c )) + ( h + g )(¯ b ( a + d ) − e ) + 2(¯ a − d )( h ( g ¯ b − el ) − e ¯ c ) + 2(¯ a + d ) e f ,(cid:15) = − (cid:15) = ¯ b ((¯ a − d )( f + c ) + ( h + g ) ( d − ¯ a )) + 2( h ( el − g ¯ b ) + 2 e ¯ c )( d − ¯ a ) − e f (¯ a + d ) . • C : α = β = e (¯ c − f ) − l ( h + g ) , γ = ( h + g ) + ( d + ¯ a )( f − ¯ c ) , δ = − (cid:15) = l ( d + ¯ a ) − e ( h + g ) . • C and C : α = β = (¯ bd − e )( f − ¯ c ) + ( l − ¯ bf )( h + g ) + 2(¯ b ¯ c − l ) gh + 2 l ( dl − e ( g + h ))( f + ¯ c ) ,β = α = (¯ a ¯ b − e )(¯ c − f ) + ( l − ¯ b ¯ c )( h + g ) + 2(¯ bf − l ) gh − l (¯ al − e ( g + h ))( f + ¯ c ) ,γ = γ = e ( h − g ) (¯ c − f ) − l ( h + g − gh ( h + g )) + e (¯ a − d )( f − ¯ c ) + l ( g + h )(¯ a − d )( f + ¯ c ) ,δ = − δ = ( h − g )((2 l + ¯ b ( f − ¯ c ))(¯ a + d ) + 2 e (¯ c − f ) + ¯ b ( h − g ) + 4 l ( h + g )) ,(cid:15) = − (cid:15) = (¯ b ( h − g ) − el )( h − g + (¯ a − d )( f + ¯ c )) , M The components of A are defined by (116); we denote ¯ a = a − ω , ¯ f = f − ω , ¯ g = g − ω and ¯ d = d − ω . All thecomponents must be divided by (cid:107) w k (cid:107) = ( α k + β k + γ k + δ k + (cid:15) k + θ k ) / , k = 1 , . . . w and w : α = − α = − (((2 l − ¯ gh − ¯ f ¯ g )( p − n ) + 2( ¯ d + e )( h − ¯ f ) l + ¯ g ( ¯ d + e )( ¯ f − h )) q + c ( − lp + ( ln + hm + ¯ fm ) p + ( ln − mn ( h + ¯ f ) + ( ¯ d + e )( ¯ f − h ) l ) p − ln + ( hm + ¯ fm ) n + (( − ¯ d − e ) h + ( ¯ d + e ) ¯ f ) ln + ( ¯ d + e ) h m + ( − ¯ d − e ) ¯ f m ) + b (¯ gp + ( − ¯ gn − lm ) p + ( − ¯ gn + 4 lmn + ( ¯ d + e )¯ gh + ( − ¯ d − e ) ¯ f ¯ g ) p + ¯ gn − lmn + (( ¯ d + e )¯ gh + ( − ¯ d − e ) ¯ f ¯ g ) n + l (( − d − e ) hm + (2 ¯ d + 2 e ) ¯ fm ))) ,β = − γ = − (((2 l − ¯ gh − ¯ f ¯ g ) p + ( − l + ¯ gh + ¯ f ¯ g ) n ) q + ( c ( − lp + (2 ln + 2 hm + 2 ¯ fm ) p + ln + ( − hm − fm ) n + (( − ¯ d − e ) h + ( ¯ d + e ) ¯ f ) l ) + b (3¯ gp + ( − gn − lm ) p − ¯ gn + 4 lmn + ( ¯ d + e )¯ gh + ( − ¯ d − e ) ¯ f ¯ g )) q + ¯ a ( − ¯ gp + 3 lmp + (¯ gn − lmn + ¯ f ( ¯ d ¯ g − m ) + h ( − m − e ¯ g ) + ( e − ¯ d ) l ) p − lmn + ( h ( m − ¯ d ¯ g ) + ¯ f ( m + e ¯ g )+ ( ¯ d − e ) l ) n + l (( ¯ d + e ) hm + ( − ¯ d − e ) ¯ fm )) + c ( p + ( − n + eh − ¯ d ¯ f ) p + ( ¯ dh − e ¯ f ) n ) + bc ( − mp + (2 mn + (2 ¯ d − e ) l ) p + mn + (2 e − d ) ln + ( − ¯ d − e ) hm + ( ¯ d + e ) ¯ fm ) + b ((2 m + ( e − ¯ d )¯ g ) p + (( ¯ d − e )¯ g − m ) n )) ,γ = − β = (((2 l − ¯ gh − ¯ f ¯ g ) p + ( − l + ¯ gh + ¯ f ¯ g ) n ) q + ( c ( − lp + ( − ln + 2 hm + 2 ¯ fm ) p + 3 ln + ( − hm − fm ) n + (( ¯ d + e ) h + ( − ¯ d − e ) ¯ f ) l ) + b (¯ gp + (2¯ gn − lm ) p − gn + 4 lmn + ( − ¯ d − e )¯ gh + ( ¯ d + e ) ¯ f ¯ g )) q + ¯ a (( lm − ¯ gn ) p + (2 lmn + h ( ¯ d ¯ g − m ) + ¯ f ( − m − e ¯ g ) + ( e − ¯ d ) l ) p + ¯ gn − lmn + ( ¯ f ( m − ¯ d ¯ g ) + h ( m + e ¯ g ) + ( ¯ d − e ) l ) n + l (( − ¯ d − e ) hm + ( ¯ d + e ) ¯ fm )) + c ( np + ( e ¯ f − ¯ dh ) p − n + ( ¯ d ¯ f − eh ) n ) + bc ( − mp + ((2 ¯ d − e ) l − mn ) p + 3 mn + (2 e − d ) ln + ( ¯ d + e ) hm + ( − ¯ d − e ) ¯ fm ) + b ((2 m + ( e − ¯ d )¯ g ) p + (( ¯ d − e )¯ g − m ) n )) ,δ = − δ = − (( b ( − lp + 4 lnp − ln + (( − d − e ) h + (2 ¯ d + 2 e ) ¯ f ) l ) + c (( h + ¯ f ) p + ( − h − f ) np + ( h + ¯ f ) n + ( ¯ d + e ) h + ( − ¯ d − e ) ¯ f )) q + ¯ a ( lp + ( − ln − hm − ¯ fm ) p + ( − ln + (2 hm + 2 ¯ fm ) n + (( ¯ d + e ) h + ( − ¯ d − e ) ¯ f ) l ) p + ln + ( − hm − ¯ fm ) n + (( ¯ d + e ) h + ( − ¯ d − e ) ¯ f ) ln + ( − ¯ d − e ) h m + ( ¯ d + e ) ¯ f m ) + bc ( − p + np + ( n + ( − ¯ d − e ) h + ( ¯ d + e ) ¯ f ) p − n + (( − ¯ d − e ) h + ( ¯ d + e ) ¯ f ) n ) + b (2 mp − mnp + 2 mn + (2 ¯ d + 2 e ) hm + ( − d − e ) ¯ fm )) ,(cid:15) = θ = (((2 h − f ) l − ¯ gh + ¯ f ¯ g ) q + ( c ((2 ¯ f − h ) lp + (2 ¯ f − h ) ln + 2 h m − f m ) + b ((2¯ gh − f ¯ g ) p + (2¯ gh − f ¯ g ) n + l (4 ¯ fm − hm ))) q + ¯ a (( l − ¯ gh ) p + ((2 ¯ f ¯ g − l ) n + l (2 hm − fm )) p + ( l − ¯ gh ) n + l (2 hm − fm ) n + ¯ f ( m + e ¯ g )+ h ( − m − e ¯ g ) + (2 eh − e ¯ f ) l ) + bc ( − lp + (4 ln − hm + 2 ¯ fm ) p − ln + (2 ¯ fm − hm ) n + (4 e ¯ f − eh ) l )+ c ( hp − fnp + hn + eh − e ¯ f ) + b (¯ gp − gnp + ¯ gn + h (2 m + 2 e ¯ g ) + ¯ f ( − m − e ¯ g ))) ,θ = (cid:15) = (((2 h − f ) l − ¯ gh + ¯ f ¯ g ) q + ( c ((2 ¯ f − h ) lp + (2 ¯ f − h ) ln + 2 h m − f m ) + b ((2¯ gh − f ¯ g ) p + (2¯ gh − f ¯ g ) n + l (4 ¯ fm − hm ))) q + ¯ a (( ¯ f ¯ g − l ) p + ((2 l − gh ) n + l (2 hm − fm )) p + ( ¯ f ¯ g − l ) n + l (2 hm − fm ) n + ¯ f ( m − ¯ d ¯ g )+ h ( ¯ d ¯ g − m ) + (2 ¯ d ¯ f − dh ) l ) + bc (2 lp + ( − ln − hm + 2 ¯ fm ) p + 2 ln + (2 ¯ fm − hm ) n + (4 ¯ dh − d ¯ f ) l )+ b ( − ¯ gp + 2¯ gnp − ¯ gn + h (2 m − d ¯ g ) + ¯ f (2 ¯ d ¯ g − m )) + c ( − ¯ fp + 2 hnp − ¯ fn − ¯ dh + ¯ d ¯ f )) ; • w : α = 2( ql + bm )( p + n ) − c ( p + n ) + ( c ( ¯ d − e ) − qm )( ¯ f + h ) − bl ( ¯ d − e ) ,β = γ = ( qc − ¯ am )( p + n ) + 2 q ( bm − lq ) + (¯ al + bc )( e − ¯ d ) ,δ = 2 q ( h + ¯ f ) − bq ( p + n ) + ¯ a ( p + n ) + ¯ a ( e − ¯ d )( h + ¯ f ) + 2 b ( ¯ d − e ) ,(cid:15) = − θ = (¯ am − qc )( ¯ f + h ) + ( bc − ¯ al )( p + n ) − b ( bm + ql ) . w and w : α = α = − ((¯ gp + ( − ¯ gn − lm ) p + ( − ¯ gn + 4 lmn + ( ¯ d + e )¯ gh + ( − ¯ d − e ) ¯ f ¯ g ) p + ¯ gn − lmn + (( ¯ d + e )¯ gh + ( − ¯ d − e ) ¯ f ¯ g ) n + l (( − d − e ) hm + (2 ¯ d + 2 e ) ¯ fm )) q + c ( − mp + ( mn + ( ¯ d − e ) l ) p + ( mn + (2 e − d ) ln + ( − ¯ d − e ) hm + ( ¯ d + e ) ¯ fm ) p − mn + ( ¯ d − e ) ln + (( − ¯ d − e ) hm + ( ¯ d + e ) ¯ fm ) n + (( ¯ d − e ) h + ( e − ¯ d ) ¯ f ) l ) + b ((2 m + ( e − ¯ d )¯ g ) p + ((2 ¯ d − e )¯ g − m ) np + (2 m + ( e − ¯ d )¯ g ) n + h ((2 ¯ d + 2 e ) m + ( e − ¯ d )¯ g ) + ¯ f (( − d − e ) m + ( ¯ d − e )¯ g ))) , + ( ¯ d − e ) ln + ( ¯ d + e )( ¯ f − h ) mn + (( ¯ d − e )( h − ¯ f )) l ) + b ((2 m + ( e − ¯ d )¯ g ) p + ((2 ¯ d − e )¯ g − m ) np + (2 m + ( e − ¯ d )¯ g ) n + h ((2 ¯ d + 2 e ) m + ( e − ¯ d )¯ g ) + ¯ f (( − d − e ) m + ( ¯ d − e )¯ g ))) ,β = γ = (¯ gq ( p − n ) + 2( ¯ d + e )(¯ gh − l ) q + ( c ( − mp + (4 mn + (2 ¯ d + 2 e ) l ) p − mn + (2 ¯ d + 2 e ) ln + ( − d − e ) hm ) + b (( − d − e )¯ gp + ( − d − e )¯ gn + (4 ¯ d + 4 e ) lm )) q + ¯ a (( m + e ¯ g ) p + ((2 ¯ d ¯ g − m ) n + ( − d − e ) lm ) p + ( m + e ¯ g ) n + ( − d − e ) lmn + h ((2 ¯ d + 2 e ) m + ( e − ¯ d )¯ g ) + ( ¯ d − e ) l ) + c ( − ep − dnp − en + ( ¯ d − e ) h )+ bc ((2 ¯ d + 2 e ) mp + (2 ¯ d + 2 e ) mn + (2 e − d ) l ) + b (( − d − e ) m + ( ¯ d − e )¯ g ) ,γ = β = (¯ gq ( p − n ) + 2( ¯ d + e )( l − ¯ f ¯ g ) q + ( c ( − mp + (4 mn + ( − d − e ) l ) p − mn + ( − d − e ) ln + (4 ¯ d + 4 e ) ¯ fm )+ b ((2 ¯ d + 2 e )¯ gp + (2 ¯ d + 2 e )¯ gn + ( − d − e ) lm )) q + ¯ a (( m − ¯ d ¯ g ) p + (( − m − e ¯ g ) n + (2 ¯ d + 2 e ) lm ) p + ( m − ¯ d ¯ g ) n + (2 ¯ d + 2 e ) lmn + ¯ f (( − d − e ) m + ( ¯ d − e )¯ g ) + ( e − ¯ d ) l ) + c ( ¯ dp + 2 enp + ¯ dn + ( e − ¯ d ) ¯ f )+ bc (( − d − e ) mp + ( − d − e ) mn + (2 ¯ d − e ) l ) + b ((2 ¯ d + 2 e ) m + ( e − ¯ d )¯ g ) ,δ = δ − (2 l ( p − n ) q + 2( ¯ d + e )( h − ¯ f ) lq + ( c ( − p + np + ( n + ( − ¯ d − e ) h + ( ¯ d + e ) ¯ f ) p − n + (( − ¯ d − e ) h + ( ¯ d + e ) ¯ f ) n ) + b ( − mp + 4 mnp − mn + ( − d − e ) hm + (2 ¯ d + 2 e ) ¯ fm )) q + ¯ a ( mp + (( e − ¯ d ) l − mn ) p + ( − mn + (2 ¯ d − e ) ln + ( ¯ d + e ) hm + ( − ¯ d − e ) ¯ fm ) p + mn + ( e − ¯ d ) ln + (( ¯ d + e ) hm + ( − ¯ d − e ) ¯ fm ) n + (( e − ¯ d ) h + ( ¯ d − e ) ¯ f ) l ) + bc (( ¯ d − e ) p + (2 e − d ) np + ( ¯ d − e ) n + ( ¯ d − e ) h + ( e − ¯ d ) ¯ f )) ,(cid:15) = − θ = β ,θ = − (cid:15) = − γ ; • w : α = ¯ g ( p + n ) − lm ( p + n ) + ( h + ¯ f )(2 m + ( e − ¯ d )¯ g ) + 2 l ( ¯ d − e ) ,β = γ = l (2 mq + c ( e − ¯ d )) + ( p + n )( cm − ¯ gq ) + b (( ¯ d − e )¯ g − m ) ,δ = − ( f + h )(2 mq + c ( e − ¯ d )) − p + n )( lq + bm ) + c ( p + n ) + 2 bl ( ¯ d − e ) ,(cid:15) = − θ = ( h + ¯ f )(¯ gq − cm ) + ( cl − b ¯ g )( p + n ) + 2 l ( bm − lq ) , M The components of A are defined by (121); we denote ¯ a = a − ω , ¯ e = e − ω , ¯ f = f − ω . All the components mustbe divided by (cid:107) w k (cid:107) = ( α k + β k + γ k + δ k + (cid:15) k + θ k ) / , k = 7 , . . . w α = (¯ ehn + ( − ghm + c (2 h + ¯ e ¯ f ) − ¯ edg ) n + ( ¯ fhm + ( d (2 g − h ) − c ¯ fg ) m + ¯ a ( h + ( g − ¯ e ¯ f ) h ) + b ( − gh − g + ¯ e ¯ fg ) − cdgh + 3 c ¯ fh + ¯ ed h ) n + ( c ¯ f − d ¯ fg ) m + ( b ( ¯ fh + ¯ fg − ¯ e ¯ f ) + 2 d gh − cd ¯ fh ) m + d ( b ((¯ e ¯ f − g ) h − h )+ ¯ a ( − gh − g + ¯ e ¯ fg ) − c ¯ fg ) + ¯ ac ( ¯ fh + ¯ fg − ¯ e ¯ f ) + cd (2 g + ¯ e ¯ f ) − ¯ ed g + c ¯ f ) ,β = (¯ egn + (( − g − ¯ e ¯ f ) m + 2 cgh + ¯ edh ) n + (3 ¯ fgm + ( − dgh − c ¯ fh ) m + b ((¯ e ¯ f − g ) h − h ) + ¯ a ( gh + g − ¯ e ¯ fg )+ cd (2 h − g ) + c ¯ fg + ¯ ed g ) n − ¯ f m + 3 d ¯ fhm + (¯ a ( − ¯ fh − ¯ fg + ¯ e ¯ f ) + d ( − h − ¯ e ¯ f ) + 2 cd ¯ fg − c ¯ f ) m + d (¯ a ( h + ( g − ¯ e ¯ f ) h ) + b ( gh + g − ¯ e ¯ fg ) + c ¯ fh ) + bc ( − ¯ fh − ¯ fg + ¯ e ¯ f ) − cd gh + ¯ ed h ) ,γ = − ( b ( h − g ) n + (2 b ¯ fgm − bdgh + 2 bc ¯ fh ) n − b ¯ f m + 2 bd ¯ fhm + bd ( g − h ) − bcd ¯ fg + bc ¯ f ) ,δ = − (¯ emn + ( − gm + 2 chm − ¯ ebh − ¯ e ¯ ag + ¯ ecd ) n + ( ¯ fm − dhm + (¯ a ( h + 3 g − ¯ e ¯ f ) − cdg + c ¯ f + ¯ ed ) m + c ( b ( − h − g − ¯ e ¯ f ) − agh ) + d (2 c h + 2¯ ebg )) n + ( b ¯ fh − ¯ a ¯ fg + cd ¯ f ) m + ( d ( b ( − h − g − ¯ e ¯ f ) + 2¯ agh ) − cd h + 2 bc ¯ fg ) m + d (¯ ac ( h + 3 g − ¯ e ¯ f ) + c ¯ f ) + b ( gh + g − ¯ e ¯ fg ) + ¯ a ( − gh − g + ¯ e ¯ fg ) + c ( − b ¯ fh − ¯ a ¯ fg ) + d (¯ ebh − c g − ¯ e ¯ ag ) + ¯ ecd ) ,(cid:15) = − (¯ ecn + (( − cg − ¯ ed ) m + 2 c h + ¯ e ¯ ah + ¯ ebg ) n + ((2 dg + c ¯ f ) m + ( b ( − h − g − ¯ e ¯ f ) − agh − cdh ) m + ¯ ac (3 h + g − ¯ e ¯ f )+ d (2¯ ebh − c g ) + c ¯ f + ¯ ecd ) n − d ¯ fm + (¯ a ¯ fh + 2 d h + b ¯ fg ) m + ( d (¯ a ( − h − g + ¯ e ¯ f ) − c ¯ f ) − bc ¯ fh + 2 cd g − ¯ ed ) m + ¯ a ( h + ( g − ¯ e ¯ f ) h ) + b ((¯ e ¯ f − g ) h − h ) + cd ( b ( h + g + ¯ e ¯ f ) − agh ) + c (¯ a ¯ fh − b ¯ fg ) + d (¯ e ¯ ah − ¯ ebg )) ,θ = (¯ en + (2 ch − gm ) n + ( ¯ fm − dhm + ¯ a ( h + g − e ¯ f ) − bgh − cdg + c ¯ f + 2¯ ed ) n + ((2 b ¯ fh + 2¯ a ¯ fg − d g ) m + bd (2 g − h ) + c ( − a ¯ fh − b ¯ fg ) + 2 cd h ) n + ( d ¯ f − ¯ a ¯ f ) m + ( d (2¯ a ¯ fh − b ¯ fg ) − d h + 2 bc ¯ f ) m + d (¯ a ( h + g − e ¯ f )+ 2 bgh + c ¯ f ) + b ( ¯ fh + ¯ fg − ¯ e ¯ f ) + ¯ a ( − ¯ fh − ¯ fg + ¯ e ¯ f ) + cd (2¯ a ¯ fg − b ¯ fh ) − cd g − ¯ ac ¯ f + ¯ ed ) ; • w α = (¯ e n + (¯ ech − egm ) n + (( − h + 2 g + ¯ e ¯ f ) m − cghm + ¯ a (¯ eh + ¯ eg − ¯ e ¯ f ) + c (¯ e ¯ f − h ) − ebgh − ecdg + ¯ e d ) n − ¯ fgm + (2 dgh − c ¯ fh ) m + ( b (( g + ¯ e ¯ f ) h − h ) + ¯ a ( gh − g + ¯ e ¯ fg ) + 2 cdg − c ¯ fg − ¯ ed g ) m + c (¯ a (( g + ¯ e ¯ f ) h − h )+ b ( − gh + g − ¯ e ¯ fg )) + d ( b ( − ¯ eh − ¯ eg + ¯ e ¯ f ) − e ¯ agh ) − c ¯ fh + ¯ ecd h ) ,β = − ((¯ ehm + ¯ ecg − ¯ e d ) n + ((2¯ edg − cg ) m + b (¯ eh + ¯ eg − ¯ e ¯ f ) + 2 c gh − e ¯ agh ) n − ¯ fhm + ( d (2 h − ¯ e ¯ f ) + c ¯ fg ) m + (¯ a (( g + ¯ e ¯ f ) h − h ) + b ( gh − g + ¯ e ¯ fg ) − cdgh − c ¯ fh + ¯ ed h ) m + c ( b (( g + ¯ e ¯ f ) h − h ) + ¯ a ( − gh + g − ¯ e ¯ fg ))+ d (¯ a ( − ¯ eh − ¯ eg + ¯ e ¯ f ) + c (2 h − g − ¯ e ¯ f ) − ebgh ) + c ¯ fg + 3¯ ecd g − ¯ e d ) ,γ = − (¯ emn + ( − gm + ¯ ebh − ¯ e ¯ ag + ¯ ecd ) n + ( ¯ fm + (¯ a ( − h + 3 g − ¯ e ¯ f ) − bgh − cdg + c ¯ f + ¯ ed ) m + bc ( − h + g + ¯ e ¯ f ) − ebdg ) n + ( b ¯ fh − ¯ a ¯ fg + cd ¯ f ) m + ( bd ( − h + g + ¯ e ¯ f ) − bc ¯ fg ) m + ¯ a ( gh − g + ¯ e ¯ fg ) + b ( − gh + g − ¯ e ¯ fg )+ d ( c (¯ a ( − h + 3 g − ¯ e ¯ f ) + 2 bgh ) + c ¯ f ) + c ( − b ¯ fh − ¯ a ¯ fg ) + d ( − ¯ ebh − c g − ¯ e ¯ ag ) + ¯ ecd ) ,δ = (¯ e bn + (2 hm + (2 c h − e ¯ ah − ebg ) m ) n + ( b ( h + g ) − agh + 2 cdh ) m + c ( b ( − h − g ) − agh )+ d ( c (2¯ ebg − e ¯ ah ) + 2 c h ) − eb gh + 2¯ e ¯ a gh − ¯ e bd ) ,(cid:15) = ((¯ em + ¯ ec − ¯ e ¯ a ) n + ( − gm + 2 chm + (2¯ ebh − c g + 2¯ e ¯ ag ) m + c (2¯ ebg − e ¯ ah ) + 2 c h − e bd ) n + ¯ fm − dhm + (¯ a ( h + g − e ¯ f ) − bgh − cdg + 2 c ¯ f + ¯ ed ) m + ( bc (2 h − g ) + d ( − c h + 2¯ e ¯ ah + 2¯ ebg )) m + c (¯ a ( h + g − e ¯ f ) + 2 bgh )+ b (¯ eh + ¯ eg − ¯ e ¯ f ) + ¯ a ( − ¯ eh − ¯ eg + ¯ e ¯ f ) + d ( c (2¯ e ¯ ag − ebh ) − c g ) + c ¯ f + (¯ ec − ¯ e ¯ a ) d ) ,θ = − (¯ ecn + (2 hm + ( − cg − ¯ ed ) m + 2 c h − ¯ e ¯ ah + ¯ ebg ) n + ((2 dg + c ¯ f ) m + ( b (3 h − g − ¯ e ¯ f ) − agh ) m + ¯ ac ( − h + g − ¯ e ¯ f ) + d ( − ebh − c g ) + c ¯ f + ¯ ecd ) n − d ¯ fm + ( − ¯ a ¯ fh + 2 d h + b ¯ fg ) m + ( d (¯ a ( h − g + ¯ e ¯ f ) − c ¯ f )+ 2 bc ¯ fh + 2 cd g − ¯ ed ) m + b ( h + ( − g − ¯ e ¯ f ) h ) + ¯ a (( g + ¯ e ¯ f ) h − h ) + cd ( b ( − h + g + ¯ e ¯ f ) − agh )+ c ( − ¯ a ¯ fh − b ¯ fg ) + d (2 c h − ¯ e ¯ ah − ¯ ebg )) ; w α = − ((¯ ehm + ¯ ecg − ¯ e d ) n + ( − ghm + ( c (2 h − g ) + 2¯ edg ) m + b ( − ¯ eh − ¯ eg + ¯ e ¯ f ) + 2 c gh − ecdh ) n + ¯ fhm + ( d ( − h − ¯ e ¯ f ) + c ¯ fg ) m + (¯ a ( h + ( g − ¯ e ¯ f ) h ) + b ( gh + g − ¯ e ¯ fg ) − cdgh + c ¯ fh + 3¯ ed h ) m + c ( b ((¯ e ¯ f − g ) h − h )+ ¯ a ( gh + g − ¯ e ¯ fg )) + d (¯ a ( − ¯ eh − ¯ eg + ¯ e ¯ f ) + c ( − g − ¯ e ¯ f )) + c ¯ fg + 3¯ ecd g − ¯ e d ) ,β = − (¯ e n + (3¯ ech − egm ) n + ((2 g + ¯ e ¯ f ) m + ( − cgh − edh ) m + ¯ a (¯ eh + ¯ eg − ¯ e ¯ f ) + c (2 h + ¯ e ¯ f ) − ecdg + ¯ e d ) n − ¯ fgm + (2 dgh + c ¯ fh ) m + ( b ((¯ e ¯ f − g ) h − h ) + ¯ a ( − gh − g + ¯ e ¯ fg ) + cd (2 g − h ) − c ¯ fg − ¯ ed g ) m + c (¯ a ( h + ( g − ¯ e ¯ f ) h ) + b ( − gh − g + ¯ e ¯ fg )) + d ( b (¯ eh + ¯ eg − ¯ e ¯ f ) − c gh ) + c ¯ fh + ¯ ecd h ) ,γ = (¯ ecn + (( − cg − ¯ ed ) m + 2 c h + ¯ e ¯ ah − ¯ ebg ) n + ((2 dg + c ¯ f ) m + ( b ( h + g + ¯ e ¯ f ) − agh − cdh ) m + ¯ ac (3 h + g − ¯ e ¯ f )+ d ( − ebh − c g ) + c ¯ f + ¯ ecd ) n − d ¯ fm + (¯ a ¯ fh + 2 d h − b ¯ fg ) m + ( d (¯ a ( − h − g + ¯ e ¯ f ) − c ¯ f ) + 2 bc ¯ fh + 2 cd g − ¯ ed ) m + ¯ a ( h + ( g − ¯ e ¯ f ) h ) + b ((¯ e ¯ f − g ) h − h ) + cd ( b ( − h − g − ¯ e ¯ f ) − agh ) + c (¯ a ¯ fh + b ¯ fg ) + d (¯ e ¯ ah + ¯ ebg )) ,δ = ((¯ em + ¯ ec − ¯ e ¯ a ) n + ( − gm + 2 chm + ( − ebh − c g + 2¯ e ¯ ag ) m + c ( − e ¯ ah − ebg ) + 2 c h + 2¯ e bd ) n + ¯ fm − dhm + (¯ a ( h + g − e ¯ f ) + 2 bgh − cdg + 2 c ¯ f + ¯ ed ) m + ( bc (2 g − h ) + d ( − c h + 2¯ e ¯ ah − ebg )) m + c (¯ a ( h + g − e ¯ f ) − bgh ) + b (¯ eh + ¯ eg − ¯ e ¯ f ) + ¯ a ( − ¯ eh − ¯ eg + ¯ e ¯ f ) + d ( c (2¯ ebh + 2¯ e ¯ ag ) − c g ) + c ¯ f + (¯ ec − ¯ e ¯ a ) d ) ,(cid:15) = (¯ e bn + (2¯ ebch − ebgm ) n + b ( g − h ) m + (2¯ ebdh − bcgh ) m + bc ( h − g ) + 2¯ ebcdg − ¯ e bd ) ,θ = − (¯ emn + ( − gm + 2 chm − ¯ ebh − ¯ e ¯ ag + ¯ ecd ) n + ( ¯ fm − dhm + (¯ a ( h + 3 g − ¯ e ¯ f ) − cdg + c ¯ f + ¯ ed ) m + c ( b ( − h − g − ¯ e ¯ f ) − agh ) + d (2 c h + 2¯ ebg )) n + ( b ¯ fh − ¯ a ¯ fg + cd ¯ f ) m + ( d ( b ( − h − g − ¯ e ¯ f ) + 2¯ agh ) − cd h + 2 bc ¯ fg ) m + d (¯ ac ( h + 3 g − ¯ e ¯ f ) + c ¯ f ) + b ( gh + g − ¯ e ¯ fg ) + ¯ a ( − gh − g + ¯ e ¯ fg ) + c ( − b ¯ fh − ¯ a ¯ fg ) + d (¯ ebh − c g − ¯ e ¯ ag ) + ¯ ecd ) ; • w α = − (¯ egn + (( − h − g − ¯ e ¯ f ) m + ¯ edh ) n + (3 ¯ fgm − c ¯ fhm + b (( − g − ¯ e ¯ f ) h − h ) + ¯ a ( gh + g − ¯ e ¯ fg )+ cd ( − h − g ) + c ¯ fg + ¯ ed g ) n − ¯ f m + d ¯ fhm + (¯ a ( ¯ fh − ¯ fg + ¯ e ¯ f ) + 2 b ¯ fgh + 2 cd ¯ fg − c ¯ f − ¯ ed ¯ f ) m + d (¯ a (( − g − ¯ e ¯ f ) h − h ) + b ( − gh − g + ¯ e ¯ fg ) − c ¯ fh ) + c ( b ( − ¯ fh + ¯ fg − ¯ e ¯ f ) + 2¯ a ¯ fgh ) + ¯ ed h ) ,β = (¯ ehn + (¯ ec ¯ f − ¯ edg ) n + ( − ¯ fhm + ( d (2 h + 2 g ) − c ¯ fg ) m + ¯ a (( − g − ¯ e ¯ f ) h − h ) + b ( gh + g − ¯ e ¯ fg )+ c ¯ fh + ¯ ed h ) n + ( c ¯ f − d ¯ fg ) m + ( b ( ¯ fh − ¯ fg + ¯ e ¯ f ) + 2¯ a ¯ fgh − cd ¯ fh ) m + d ( b (( − g − ¯ e ¯ f ) h − h ) + ¯ a ( − gh − g + ¯ e ¯ fg ) − c ¯ fg ) + c (¯ a ( − ¯ fh + ¯ fg − ¯ e ¯ f ) + 2 b ¯ fgh ) + cd (2 h + 2 g + ¯ e ¯ f ) − ¯ ed g + c ¯ f ) ,γ = (¯ en − gmn + ( ¯ fm + ¯ a ( − h + g − e ¯ f ) − cdg + c ¯ f + 2¯ ed ) n + ((2¯ a ¯ fg − d g ) m + bd ( − h − g ) + 2 bc ¯ fg ) n + ( d ¯ f − ¯ a ¯ f ) m + (2 bd ¯ fg − bc ¯ f ) m + ¯ a ( ¯ fh − ¯ fg + ¯ e ¯ f ) + b ( − ¯ fh + ¯ fg − ¯ e ¯ f ) + d (¯ a ( − h + g − e ¯ f ) + c ¯ f )+ 2¯ acd ¯ fg − cd g − ¯ ac ¯ f + ¯ ed ) ,δ = (¯ ecn + ( − hm + ( − cg − ¯ ed ) m + ¯ e ¯ ah − ¯ ebg ) n + ((2 dg + c ¯ f ) m + ( b ( h + g + ¯ e ¯ f ) + 2¯ agh − cdh ) m + ¯ ac ( − h + g − ¯ e ¯ f ) + d ( − ebh − c g ) + c ¯ f + ¯ ecd ) n − d ¯ fm + (¯ a ¯ fh − b ¯ fg ) m + ( d (¯ a ( h − g + ¯ e ¯ f ) − c ¯ f )+ 2 bc ¯ fh + 2 cd g − ¯ ed ) m + b ( h + ( g + ¯ e ¯ f ) h ) + ¯ a (( − g − ¯ e ¯ f ) h − h ) + cd ( b ( − h − g − ¯ e ¯ f ) + 2¯ agh )+ c (¯ a ¯ fh + b ¯ fg ) + d ( − c h + ¯ e ¯ ah + ¯ ebg )) ,(cid:15) = − (¯ emn + ( − gm + 2 chm + ¯ ebh − ¯ e ¯ ag + ¯ ecd ) n + ( ¯ fm − dhm + (¯ a ( h + 3 g − ¯ e ¯ f ) − cdg + c ¯ f + ¯ ed ) m + c ( b ( h + g + ¯ e ¯ f ) − agh ) + d (2 c h − ebg )) n + ( − b ¯ fh − ¯ a ¯ fg + cd ¯ f ) m + ( d ( b ( h + g + ¯ e ¯ f ) + 2¯ agh ) − cd h − bc ¯ fg ) m + d (¯ ac ( h + 3 g − ¯ e ¯ f ) + c ¯ f ) + b ( gh + g − ¯ e ¯ fg ) + ¯ a ( − gh − g + ¯ e ¯ fg ) + c ( b ¯ fh − ¯ a ¯ fg ) + d ( − ¯ ebh − c g − ¯ e ¯ ag ) + ¯ ecd ) ,θ = (2 hmn + ( b ( h + g ) − agh + 2 cdh ) n + ( − a ¯ fh + 2 d h − b ¯ fg ) mn + b ¯ f m + d ( b ( − h − g ) − agh )+ cd (2 b ¯ fg − a ¯ fh ) − b ¯ fgh + 2¯ a ¯ fgh + 2 cd h − bc ¯ f ) ; w α = ((2 h m + ¯ a ( gh + g − ¯ e ¯ fg ) + c ( gh − g + ¯ e ¯ fg ) − edh ) n + ( c ( ¯ fh + ¯ fg − ¯ e ¯ f ) + ¯ a ( ¯ fh − ¯ fg + ¯ e ¯ f ) − dgh ) m + b ( h − g + 2¯ e ¯ fg − ¯ e ¯ f ) + d ( c ( h + ( − g − ¯ e ¯ f ) h ) + ¯ a (( − g − ¯ e ¯ f ) h − h )) + 2¯ ac ¯ fgh + 2¯ ed gh ) ,β = ((¯ eh + ¯ eg − ¯ e ¯ f ) n + ((2¯ e ¯ fg − g ) m + 2 cg h − edgh ) n + ( − ¯ fh + ¯ fg − ¯ e ¯ f ) m + ( d (2 h + 2¯ e ¯ fh ) − c ¯ fgh ) m + ¯ a ( − h + g − e ¯ fg + ¯ e ¯ f ) + c ( ¯ fh + ¯ fg − ¯ e ¯ f ) + d ( − ¯ eh + ¯ eg − ¯ e ¯ f ) + cd (2¯ e ¯ fg − g )) ,γ = (¯ ehn + ( − ghm + cg + ¯ a (¯ e ¯ f − g ) − ¯ edg ) n + ( ¯ fhm + (2 dg − c ¯ fg ) m + ¯ a (( g − ¯ e ¯ f ) h − h ) + b ( − gh + g − ¯ e ¯ fg ) − cdgh + c ¯ fh + ¯ ed h ) n + ( c ¯ f − d ¯ fg ) m + b ( ¯ fh − ¯ fg + ¯ e ¯ f ) m + d ( b (( g − ¯ e ¯ f ) h − h ) + ¯ a ( gh − g + ¯ e ¯ fg ) − c ¯ fg − ac ¯ fg ) + d (¯ a ( h + ¯ e ¯ f ) + c (2 g − h )) + ¯ a ( − ¯ fh + ¯ fg − ¯ e ¯ f ) − ¯ ed g + ¯ ac ¯ f ) ,δ = − (¯ e n + (¯ e ¯ ah − egm ) n + ((2 h + 2 g + ¯ e ¯ f ) m + ( − cgh + ¯ agh − edh ) m + ¯ a ( − ¯ eh + ¯ eg − ¯ e ¯ f ) − ebgh + ¯ acg + c (¯ e ¯ f − g ) + d ( − ¯ ecg − ¯ e ¯ ag ) + ¯ e d ) n − ¯ fgm + c ¯ fhm + ( b ( h + ( g + ¯ e ¯ f ) h ) + d ( c ( h + 2 g − ¯ e ¯ f ) + ¯ a ( h + ¯ e ¯ f ))+ ¯ a ( − gh − g + ¯ e ¯ fg ) − ¯ ac ¯ fg − ¯ ed g ) m + ¯ a (( − g − ¯ e ¯ f ) h − h ) + d ( b ( − ¯ eh + ¯ eg − ¯ e ¯ f ) − c gh + ¯ acgh + 2¯ e ¯ agh )+ bc ( − gh − g + ¯ e ¯ fg ) + ¯ ac ¯ fh − ¯ ecd h ) ,(cid:15) = − ((¯ ehm + ¯ ecg − ¯ e d ) n + ( − ghm + ( c (2 h − g − ¯ e ¯ f ) + ¯ a (¯ e ¯ f − g ) + 2¯ edg ) m + b (¯ eh + ¯ eg − ¯ e ¯ f )+ d ( − ¯ ech − ¯ e ¯ ah ) + c gh + ¯ acgh ) n + ¯ fhm + ( d ( − h − ¯ e ¯ f ) + c ¯ fg ) m + (¯ a ( h + ( g − ¯ e ¯ f ) h )+ b ( − gh − g + ¯ e ¯ fg ) + d ( − cgh − ¯ agh ) + ¯ ac ¯ fh + 3¯ ed h ) m + bc ( h + ( g − ¯ e ¯ f ) h ) + d (¯ a ( − ¯ eh − ¯ eg + ¯ e ¯ f )+ c ( h − ¯ e ¯ f ) + ¯ ac ( − h − g )) + ¯ a ( gh + g − ¯ e ¯ fg ) + d (2¯ ecg + ¯ e ¯ ag ) + ¯ ac ¯ fg − ¯ e d ) ,θ = (¯ egn + ((2 h − g − ¯ e ¯ f ) m + cgh + ¯ agh − ¯ edh ) n + (3 ¯ fgm + ( − dgh − c ¯ fh + ¯ a ¯ fh ) m + b ( h + ( − g − ¯ e ¯ f ) h )+ d ( c ( h − g ) + ¯ a ( − h − g )) + ¯ a ( − gh + g − ¯ e ¯ fg ) + ¯ ac ¯ fg + ¯ ed g ) n − ¯ f m + d ¯ fhm + (¯ a ( − ¯ fh − ¯ fg + ¯ e ¯ f )+ d (2 h − ¯ e ¯ f ) + 2 b ¯ fgh + d ( c ¯ fg + ¯ a ¯ fg ) − ¯ ac ¯ f ) m + d (¯ a (( g + ¯ e ¯ f ) h − h ) + b ( − gh + g − ¯ e ¯ fg ) + ¯ ac ¯ fh )+ bc ( − ¯ fh − ¯ fg + ¯ e ¯ f ) + d (¯ agh − cgh ) − a ¯ fgh − ¯ ed h ) , • w α = ((¯ eh − ¯ eg + ¯ e ¯ f ) n + ((2 g − e ¯ fg ) m + c (2 h + 2¯ e ¯ fh ) − edgh ) n + ( − ¯ fh − ¯ fg + ¯ e ¯ f ) m + (2 dg h − c ¯ fgh ) m + ¯ a ( h − g + 2¯ e ¯ fg − ¯ e ¯ f ) + c ( ¯ fh − ¯ fg + ¯ e ¯ f ) + d ( − ¯ eh − ¯ eg + ¯ e ¯ f ) + cd (2 g − e ¯ fg )) ,β = (2¯ eghn + (( − g − e ¯ f ) hm + 2 cgh + 2¯ edh ) n + 2 ¯ fghm + ( − dgh − c ¯ fh ) m + b ( − h + g − e ¯ fg + ¯ e ¯ f ) + 2 cdh ) ,γ = (¯ egn + (( − g − ¯ e ¯ f ) m + ¯ edh ) n + (3 ¯ fgm − dghm + b (( g − ¯ e ¯ f ) h − h ) + ¯ a ( − gh + g − ¯ e ¯ fg ) − cdg + c ¯ fg + ¯ ed g ) n − ¯ f m + d ¯ fhm + (¯ a ( ¯ fh − ¯ fg + ¯ e ¯ f ) + 2 cd ¯ fg − c ¯ f − ¯ ed ¯ f ) m + d (¯ a (( g − ¯ e ¯ f ) h − h ) + b ( gh − g + ¯ e ¯ fg ) + c ¯ fh )+ bc ( − ¯ fh + ¯ fg − ¯ e ¯ f ) − cd gh + ¯ ed h ) ,δ = − ((¯ ehm − ¯ ecg + ¯ e d ) n + (( c (2 h + 2 g ) − edg ) m + b ( − ¯ eh + ¯ eg − ¯ e ¯ f ) − e ¯ agh + 2¯ ecdh ) n − ¯ fhm + (¯ ed ¯ f − c ¯ fg ) m + (¯ a ( h + ( g + ¯ e ¯ f ) h ) + b ( − gh − g + ¯ e ¯ fg ) − c ¯ fh − ¯ ed h ) m + c ( b (( − g − ¯ e ¯ f ) h − h )+ ¯ a ( − gh − g + ¯ e ¯ fg )) + d (¯ a ( − ¯ eh + ¯ eg − ¯ e ¯ f ) + c (2 h + 2 g + ¯ e ¯ f ) + 2¯ ebgh ) − c ¯ fg − ecd g + ¯ e d ) ,(cid:15) = − (¯ e n + (3¯ ech − egm ) n + ((2 g + ¯ e ¯ f ) m + ( − cgh − edh ) m + ¯ a (¯ eh + ¯ eg − ¯ e ¯ f ) + c (2 h + ¯ e ¯ f ) − ecdg + ¯ e d ) n − ¯ fgm + (2 dgh + c ¯ fh ) m + ( b ( h + ( g − ¯ e ¯ f ) h ) + ¯ a ( − gh − g + ¯ e ¯ fg ) + cd (2 g − h ) − c ¯ fg − ¯ ed g ) m + c (¯ a ( h + ( g − ¯ e ¯ f ) h ) + b ( gh + g − ¯ e ¯ fg )) + d ( b ( − ¯ eh − ¯ eg + ¯ e ¯ f ) − c gh ) + c ¯ fh + ¯ ecd h ) ,θ = (¯ ehn + ( c (2 h − ¯ e ¯ f ) + ¯ edg ) n + ( − ¯ fhm + (2 c ¯ fg − dg ) m + ¯ a ( h + ( − g − ¯ e ¯ f ) h ) + b ( − gh + g − ¯ e ¯ fg ) + 2 cdgh − c ¯ fh + ¯ ed h ) n + ( d ¯ fg − c ¯ f ) m + ( b ( − ¯ fh − ¯ fg + ¯ e ¯ f ) + 2¯ a ¯ fgh − d gh ) m + d ( b (( g + ¯ e ¯ f ) h − h )+ ¯ a ( − gh + g − ¯ e ¯ fg ) + 3 c ¯ fg ) + c (¯ a ( − ¯ fh − ¯ fg + ¯ e ¯ f ) − b ¯ fgh ) + cd (2 h − g − ¯ e ¯ f ) + ¯ ed g − c ¯ f ) . N The components, which are obtained from those in the subspace M by setting d = e = m = n = p = q = 0, are: • w α = c ( h + ¯ f ) − bl , β = γ = ¯ al − bc ∗ , δ = − ¯ a ( h + ¯ f ) + 2 bb ∗ , w and w α = α = ( b ¯ g − cl )( ¯ f − h ) , β = γ = − (¯ a ( l − ¯ gh ) − bc ∗ l + cc ∗ h + bb ∗ ¯ g ) ,γ = β = ¯ a ( l − ¯ f ¯ g ) − bc ∗ l + b ¯ g + cc ∗ ¯ f , δ = δ = (¯ al − bc ∗ )( ¯ f − h ) . • w α = 2 l − ¯ g ( h + ¯ f ) , β = γ = b ¯ g − cl , δ = c ( h + ¯ f ) − bl ;they must be divided by (cid:107) w k (cid:107) = ( α k + β k + γ k + δ k ) / , k = 1 , . . . N The components, which are obtained from those in the subspace M by setting m = n = h = 0: • w and w : α = β = (¯ e ¯ f − dg )(¯ a ( g − c (cid:63) ¯ f ) + d ( d (cid:63) ¯ e − c (cid:63) g ) + c ( c (cid:63) ¯ f − d (cid:63) g )) ,β = α = b ( dg − c ¯ f )( g − ¯ e ¯ f ) ,γ = θ = − b ( dg − c ¯ f ) ,δ = (cid:15) = − (( g − ¯ e ¯ f )( g ( b − ¯ a ) + ¯ acd (cid:63) ) + ( dc − ¯ ag )( c (cid:63) ( c ¯ f − dg ) + d ( d (cid:63) ¯ e − c (cid:63) g ))) ,(cid:15) = δ = b (¯ ed − cg )( gd (cid:63) − ¯ e ¯ f ) ,θ = γ = ¯ f ( b − ¯ a )( g − ¯ e ¯ f ) + ¯ a ( dg + c ¯ f ) − a ¯ f ( cc (cid:63) barf + d ¯ e ) + cdd (cid:63) ( c ¯ f − dg ) + dd (cid:63) d (¯ ed (cid:63) − c (cid:63) g ) , • w and w : α = β = b ( cg − d ¯ e )( g − ¯ e ¯ f ) ,β = α = ( d ¯ e − cg )(¯ a ( g − ¯ e ¯ f ) + d ( d ¯ e − gc ) + c (cid:63) ( c ¯ f − dg )) ,γ = θ = − (( g − ¯ e ¯ f )( g ( b − ¯ a ) + ¯ acd (cid:63) ) + ( dc − ¯ ag )( c (cid:63) ( c ¯ f − dg ) + d ( d (cid:63) ¯ e − c (cid:63) g ))) ,δ = (cid:15) = − b ( cg − ¯ ed ) ,(cid:15) = δ = ¯ e ( b − ¯ a )( g − ¯ e ¯ f ) + ¯ a ( cg + d ¯ e ) − a ¯ e ( cc (cid:63) ¯ f + dd (cid:63) ¯ e ) + dc (cid:63) c (¯ ed (cid:63) − c (cid:63) g ) − cc (cid:63) c ( d (cid:63) g − ¯ e ¯ f ) ,θ = γ = b ( dg − c ¯ f )(¯ ed (cid:63) − c (cid:63) g ) , • w and w : α = β = − b ( g − ¯ e ¯ f ) ,β = α = ( g − ¯ e ¯ f )(¯ a ( g − ¯ e ¯ f ) + c ( c (cid:63) ¯ f − d (cid:63) g ) + d (¯ ed (cid:63) − c (cid:63) g )) ,γ = θ = ( c ¯ f − dg )(¯ a ( g − ¯ e ¯ f ) + c (cid:63) ( c ¯ f − dg ) + d (cid:63) (¯ ed − cg )) ,δ = (cid:15) = b ( g − ¯ e ¯ f )( cg − ¯ ed ) ,(cid:15) = δ = (¯ ed − cg )(¯ a ( g − ¯ e ¯ f ) + c (cid:63) ( c ¯ f − dg ) + d (cid:63) (¯ ed − cg )) ,θ = γ = b ( g − ¯ e ¯ f )( dg − c ¯ f ) , also these components must be divided by (cid:107) w k (cid:107) = ( α k + β k + γ k + δ k + (cid:15) k + θ k ) / , k = 7 , . . .12.