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On Keller Theorem for Anisotropic Media Leonid G. Fel School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv 69978, Israel e-mail: [email protected] (February 1, 2008) 2 0 0 2 Abstract The EMA theory of the infinite 2D n-component isotropic comosite has its consequence the Bruggemann n a The Keller theorem in the problem of effective con- Eqn [4] J ductivity in anisotropic two-dimensional (2D) many- n σ −σ (σ ) 7 component composites makes it possible to establish a k B k =0, (4) 1 simple inequality σies(σi−1)·σies(σk) > 1 for the isotropic kX=1 σk+σB(σk) part σe(σ ) of the 2-nd rank symmetric tensor σe of ] is k i,j which necessarely leads to the duality relation n effective conductivity. n b σ (σ−1)·σ (σ )=1 (5) B k B k - Pacs: 73.50.Jt, 72.15.Gd, 72.80.Tm, 03.50.Kk s that reflects both the conformal invariance of the i d Maxwell Eqns in 2D isotropic comosite and S - n t. TheextensionoftheKellertheorem[1]intheproblem permutationinvarianceofthen-colourtessellationofthe a m ofeffectiveconductivityintheinfinite2Dtwo-component plane. The latter means that σies can satisfy the Brugge- composites on the many-componentcase [2]poses a new mann Eqn only for isotropic Sn-permutation invariant d- question on the duality relation for the 2-nd rank sym- media: σexx =σeyy, σexy =0inanyreferenceframe{x,y}. n metric tensor σe of effective conductivity in anisotropic The infinite periodic 2D three-component checker- i,j o media. It is related to the restrictions imposed on the boardwbasconsbiderebdin[3]forsymmetricallyrelatedpar- [c linear invarianbt of σie,j which is called an isotropic part tial conductivities (σ1 =1,σ2,3 =1±δ). σe(σ ) of effective conductivity. Recently the pertur- is k 1 bation theory for tbhe infinite periodic three-component v σ σ σ 2Dcheckerboardwithtwo-foldrotationlatticesymmetry 1 2 3 9 1 was developed [3] where the coincidence of σies(σk) with 3 solution σB(σk) of Bruggemann Eqn was established up σ2 σ3 σ1 1 to the 6-th order term. This fact is curious because it 0 gives grounds to think that Effective Medium Approxi- σ σ σ Principal 02 mation (EMA) describes exactly σies(σk) in this certain 3 1 2 axes structure. Here we will discuss this conclusion. / t a Let us define the isotropic part of conductivity tensor Such structure doesn’t possess an isotropy of the 2-nd m 1 rankconductivitytensorσi,j thatfollowsfromthesimple σe(σ )= Tr σe (σ ), k =1,2,...,n, (1) e - is k 2 i,j k crystallographycal consideration [5] as well as from the d straightforward calculatiobn [3] of the non-diagonal term n which is an invariantbscalar with respect to the plane o rotation and recall the Keller theorem for the princi- σexy ∝δ2. Thereforeσies(σk)forthisstructurecannotsat- c pal values σxx, σyy of diagonalized matrix σij for 2D isfythe BruggemannEqn(4)evenifits coincidencewith v: n-componentecompeosite e σbB(σk) riched the δ6 term in the perturbation theory. i b b b X σexx(σ1−1,σ2−1,...,σn−1)·σeyy(σ1,σ2,...,σn)=1, ar σbeyy(σ1−1,σ2−1,...,σn−1)·σbexx(σ1,σ2,...,σn)=1. (2) Botbh (1) and (2) make ubs possible to derive a simple inequality for Λe =σe(σ−1)·σe(σ ) is is i is k [1] J. B. Keller, J. Math. Phys. 5, 548 (1964) Λe = 1 2+σxx(σ )·σxx(σ−1)+σyy(σ )·σyy(σ−1) = [2] L. G. Fel, V. Sh. Machavariani and D. J. Bergman, J. is 4 e k e k e k e k Phys. A : Math. Gen., 33, 6669 (2000) (cid:2) (cid:3) 1 2+bσexx(σk) +b σeyy(σk) b≥1, b (3) [3] I. M. Khalatnikov and A. Kamenshchik, JETP 91, 1261 4(cid:20) σeyy(σk) σexx(σk)(cid:21) (2000) b b [4] D. A. G. Bruggemann, Ann. Physik (Leipzig) 24, 636, where the only isotropic media σxx =σyy corresponds to b b e e (1935);R.Landauer,in”ElectricalTransportandOptical the equality in (3). At the same time another isotropic Properties of Inhomogeneous Media”, Eds. J. C. Garland invariant ∆eis =detσiej(σk) satisfibes thebduality relation and D. B. Tanner, AIPConf. Proceed. No. 40, 2, (1978) [5] C. Hermann, Zs. Kristallogr. 89, 32 (1934) ∆eisb(σk)·∆eis(σk−1)=1. 1