On Keller Theorem for Anisotropic Media

1 Pages·0.06 MB·English

by  Leonid G. Fel

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
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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.
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