Tensors of domain-related equivalent structures
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In this section, TENSOR provides the symmetry-adapted form of a list of material tensors for a particular space group, as well as the tensors of the domain-related equivalent structures corresponding to a particular choice of the group of the high-symmetry phase (parent group) and the transformation between the parent group and the space group (low-symmetry group, subgroup of the parent group). On the one hand, the parent group, the space group and the transformation matrix between them must be specified. On the other hand, a tensor must be defined by the user or selected from the lists of known equilibrium, optical and transport tensors, gathered from scientific literature. If a standard space group is defined and a known tensor is selected from the lists the program will obtain the required tensor from and internal database; otherwise, the tensor is calculated live. The working setting is defined by the rules explained here. Live calculation of tensors may take too much time and even exceed the time limit, giving an empty result, if high-rank tensors, a lot of symmetry elements and/or rare settings are introduced.
Further information can be found here
Information about the selected tensor • 1 st rank Electric polarization vector Pi • Polar tensor invariant under time-reversal symmetry operation • Intrinsic symmetry symbol: V
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Information about the selected tensor • 1 st rank Electrocaloric effect tensor pi • Polar tensor invariant under time-reversal symmetry operation • Defining equation: ΔS=piEi • Electric field E Entropy variation ΔS • Intrinsic symmetry symbol: V
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Information about the selected tensor • 1 st rank Heat of polarization tensor ti • Polar tensor invariant under time-reversal symmetry operation • Defining equation: ΔS=tiΔPi • Polarization vector P variation Entropy variation ΔS • Intrinsic symmetry symbol: V
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Information about the selected tensor • 1 st rank Piezoelectric polarization tensor under hydrostatic pressure dijj • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi=-dijjp • Hydrostatic pressure p Polarization vector P • Intrinsic symmetry symbol: V
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Information about the selected tensor • 1 st rank Pyroelectric tensor pi • Polar tensor invariant under time-reversal symmetry operation • Defining equation: ΔPi=piΔT • Temperature variation ΔT Polarization vector P variation • Intrinsic symmetry symbol: V
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Information about the selected tensor • 2 nd rank Dielectric impermeability tensor βij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ei=βijDj • Electric displacement field D Electric field E • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • βij = βji
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Information about the selected tensor • 2 nd rank Dielectric permittivity tensor εij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Di=εijEj • Electric field E Electric displacement field D • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • εij = εji
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Information about the selected tensor • 2 nd rank Dielectric susceptibility tensor χeij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi=χeijEj • Electric field E Polarization vector P variation • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • χeij = χeji
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Information about the selected tensor • 2 nd rank Heat of deformation tensor αij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: ΔS=αijεij • Strain tensor εij Entropy variation ΔS • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • αij = αji
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Information about the selected tensor • 2 nd rank Magnetic permeability tensor μij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Bi=μmijHj • Magnetic field H Magnetic field B • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • μij = μji
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Information about the selected tensor • 2 nd rank Magnetic susceptibility tensor χmij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Mi=χmijHj • Magnetic field H Magnetization M • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • χmij = χmji
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Information about the selected tensor • 2 nd rank Piezocaloric effect tensor βij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: ΔS=βijσij • Stress tensor σij Entropy variation ΔS • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • βij = βji
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Information about the selected tensor • 2 nd rank Strain by hydrostatic pressure sijkk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: εij=-sijkkp • Hydrostatic pressure p Strain tensor εij • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • sij = sji
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Information about the selected tensor • 2 nd rank Strain tensor εij • Polar tensor invariant under time-reversal symmetry operation • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • εij = εji
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Information about the selected tensor • 2 nd rank Thermal expansion tensor αij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: εij=αijΔT • Temperature variation ΔT Strain tensor εij • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • αij = αji
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Information about the selected tensor • 2 nd rank Thermoelasticity tensor βij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: σij=-βijΔT • Temperature variation ΔT Stress tensor σij • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • βij = βji
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Information about the selected tensor • 2 nd rank Toroidic susceptibility tensor τij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ti=τijSj • Toroidal field S Toroidal moment T • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • τij = τji
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Information about the selected tensor • 2 nd rank Magnetotoroidic tensor ζij (direct effect) • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Mi=ζijSj • Toroidal field S Magnetization M • Intrinsic symmetry symbol: eV2
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Information about the selected tensor • 2 nd rank Magnetotoroidic tensor ζTij (inverse effect) • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Ti=ζTijHj • Magnetic field H Toroidal moment T • Intrinsic symmetry symbol: eV2
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Information about the selected tensor • 3 rd rank Acoustoelectricity tensor ρijk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: σij=ρijkJk • Alternating electric current density J Stress tensor σij • Intrinsic symmetry symbol: [V2]V • Symmetrized indexes due to intrinsic symmetry: • ρijk = ρjik
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Information about the selected tensor • 3 rd rank Isothermal piezoelectric tensor eTijk (inverse effect) • Polar tensor invariant under time-reversal symmetry operation • Defining equation: σij=eTijkEk • Electric field E Stress tensor σij • Intrinsic symmetry symbol: [V2]V • Symmetrized indexes due to intrinsic symmetry: • eTijk = eTjik
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Information about the selected tensor • 3 rd rank Piezoelectric tensor dTijk (inverse effect) • Polar tensor invariant under time-reversal symmetry operation • Defining equation: εij=dTijkEk • Electric field E Strain tensor εij • Intrinsic symmetry symbol: [V2]V • Symmetrized indexes due to intrinsic symmetry: • dTijk = dTjik
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Information about the selected tensor • 3 rd rank Isothermal piezoelectric tensor eijk (direct effect) • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Di=eijkεjk • Strain tensor εij Electric displacement field D • Intrinsic symmetry symbol: V[V2] • Symmetrized indexes due to intrinsic symmetry: • eijk = eikj
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Information about the selected tensor • 3 rd rank Piezoelectric tensor dijk (direct effect) • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi=dijkσjk • Stress tensor σij Polarization vector P • Intrinsic symmetry symbol: V[V2] • Symmetrized indexes due to intrinsic symmetry: • dijk = dikj
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Information about the selected tensor • 3 rd rank Second order magnetoelectric tensor αTijk (inverse effect) • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi=αTijkHjHk • Magnetic field H Polarization P • Intrinsic symmetry symbol: V[V2] • Symmetrized indexes due to intrinsic symmetry: • αTijk = αTikj
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Information about the selected tensor • 4 th rank Elastic compliance tensor Sijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: εij=Sijklσkl • Stress tensor σij Strain tensor εij • Intrinsic symmetry symbol: [[V2][V2]] • Symmetrized indexes due to intrinsic symmetry: • Sijkl = Sjikl • Sijkl = Sijlk • Sijkl = Sklij
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Information about the selected tensor • 4 th rank Elastic stiffness tensor Cijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: σij=Cijklεkl • Strain tensor εij Stress tensor σij • Intrinsic symmetry symbol: [[V2][V2]] • Symmetrized indexes due to intrinsic symmetry: • Cijkl = Cjikl • Cijkl = Cijlk • Cijkl = Cklij
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Information about the selected tensor • 4 th rank Viscosity tensor ηijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: σij=ηijkl∂εkl/∂t • Strain tensor rate ∂εij/∂t Stress tensor σij • Intrinsic symmetry symbol: [[V2][V2]] • Symmetrized indexes due to intrinsic symmetry: • ηijkl = ηjikl • ηijkl = ηijlk • ηijkl = ηklij
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Information about the selected tensor • 4 th rank Damage effect tensor Dijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: σ'ij=Dijklσkl • Stress tensor σij (before damage) Effective stress tensor σij' (after damage) • Intrinsic symmetry symbol: [V2][V2] • Symmetrized indexes due to intrinsic symmetry: • Dijkl = Djikl • Dijkl = Dijlk
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Information about the selected tensor • 4 th rank Electrostriction tensor γijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: εij=γijklEkEl • Electric field E and Electric field E Strain tensor εij • Intrinsic symmetry symbol: [V2][V2] • Symmetrized indexes due to intrinsic symmetry: • γijkl = γjikl • γijkl = γijlk
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Information about the selected tensor • 4 th rank Magnetostriction tensor Nijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: εij=NijklMkMl • Magnetization M and Magnetization M Strain tensor εij • Intrinsic symmetry symbol: [V2][V2] • Symmetrized indexes due to intrinsic symmetry: • Nijkl = Njikl • Nijkl = Nijlk
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Information about the selected tensor • 4 th rank Elastothermoelectric power tensor Eijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: ΔΣij=Eijklεkl • Strain tensor εij Thermoelectric power tensor variation ΔΣij • Intrinsic symmetry symbol: V2[V2] • Symmetrized indexes due to intrinsic symmetry: • Eijkl = Eijlk
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Information about the selected tensor • 4 th rank Flexoelectric tensor μijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi=μijkl∇lεjk • Strain tensor gradient ∇kεij Polarization vector P • Intrinsic symmetry symbol: V2[V2] • Symmetrized indexes due to intrinsic symmetry: • μijkl = μijlk
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Information about the selected tensor • 4 th rank Piezothermoelectric power tensor Πijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: ΔΣij=Πijklσkl • Stress tensor σij Thermoelectric power tensor variation ΔΣij • Intrinsic symmetry symbol: V2[V2] • Symmetrized indexes due to intrinsic symmetry: • Πijkl = Πijlk
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Information about the selected tensor • 5 th rank Acoustic activity tensor bijklm • Polar tensor invariant under time-reversal symmetry operation • Defining equation: σij=bijklm∇mεkl • Strain tensor gradient ∇lεij Stress tensor σij • Intrinsic symmetry symbol: [[V2][V2]]V • Symmetrized indexes due to intrinsic symmetry: • bijklm = bjiklm • bijklm = bijlkm • bijklm = bklijm
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Information about the selected tensor • 5 th rank Second-order piezoelectric tensor dijklm • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi=dijklmσjkσlm • Stress tensor σij and Stress tensor σij Polarization vector P • Intrinsic symmetry symbol: V[[V2][V2]] • Symmetrized indexes due to intrinsic symmetry: • dijklm = dikjlm • dijklm = dijkml • dijklm = dilmjk
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Information about the selected tensor • 6 th rank Third order elastic compliance tensor Sijklmn • Polar tensor invariant under time-reversal symmetry operation • Defining equation: εij=Sijklmnσklσmn • Stress tensor σij and Stress tensor σij Strain tensor εij • Intrinsic symmetry symbol: [[V2][V2][V2]] • Symmetrized indexes due to intrinsic symmetry: • Sijklmn = Sjiklmn • Sijklmn = Sijlkmn • Sijklmn = Sijklnm • Sijklmn = Sijmnkl = Sklijmn = Sklmnij = Smnijkl = Smnklij
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Information about the selected tensor • 6 th rank Third order elastic stiffness tensor Cijklmn • Polar tensor invariant under time-reversal symmetry operation • Defining equation: σij=Cijklmnεklεmn • Strain tensor εij and Strain tensor εij Stress tensor σij • Intrinsic symmetry symbol: [[V2][V2][V2]] • Symmetrized indexes due to intrinsic symmetry: • Cijklmn = Cjiklmn • Cijklmn = Cijlkmn • Cijklmn = Cijklnm • Cijklmn = Cijmnkl = Cklijmn = Cklmnij = Cmnijkl = Cmnklij
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Information about the selected tensor • 8 th rank Damage tensor Rijklmnpq • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Cijkl'=RijklmnpqCmnpq • Elastic stiffness tensor Cijkl (before damage) Elastic stiffness tensor Cijkl' (after damage) • Intrinsic symmetry symbol: [[[V2][V2]][[V2][V2]]] • Symmetrized indexes due to intrinsic symmetry: • Rijklmnpq = Rjiklmnpq • Rijklmnpq = Rijlkmnpq • Rijklmnpq = Rijklnmpq • Rijklmnpq = Rijklmnqp • Rijklmnpq = Rklijmnpq • Rijklmnpq = Rijklpqmn • Rijklmnpq = Rmnpqijkl
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Information about the selected tensor • 8 th rank Fourth order elastic compliance tensor Sijklmnpq • Polar tensor invariant under time-reversal symmetry operation • Defining equation: εij=Sijklmnpqσklσmnσpq • Stress tensor σij and Stress tensor σij and Stress tensor σij Strain tensor εij • Intrinsic symmetry symbol: [[V2][V2][V2][V2]] • Symmetrized indexes due to intrinsic symmetry: • Sijklmnpq = Sjiklmnpq • Sijklmnpq = Sijlkmnpq • Sijklmnpq = Sijklnmpq • Sijklmnpq = Sijklmnqp • Sijklmnpq = Sijklpqmn = Sijmnklpq = Sijmnpqkl = Sijpqklmn = Sijpqmnkl = Sklijmnpq = Sklijpqmn = Sklmnijpq = Sklmnpqij = Sklpqijmn = Sklpqmnij = Smnijklpq = Smnijpqkl = Smnklijpq = Smnklpqij = Smnpqijkl = Smnpqklij = Spqijklmn = Spqijmnkl = Spqklijmn = Spqklmnij = Spqmnijkl = Spqmnklij
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Information about the selected tensor • 8 th rank Fourth order elastic stiffness tensor Cijklmnpq • Polar tensor invariant under time-reversal symmetry operation • Defining equation: σij=Cijklmnpqεklεmnεpq • Strain tensor εij and Strain tensor εij and Strain tensor εij Stress tensor σij • Intrinsic symmetry symbol: [[V2][V2][V2][V2]] • Symmetrized indexes due to intrinsic symmetry: • Cijklmnpq = Cjiklmnpq • Cijklmnpq = Cijlkmnpq • Cijklmnpq = Cijklnmpq • Cijklmnpq = Cijklmnqp • Cijklmnpq = Cijklpqmn = Cijmnklpq = Cijmnpqkl = Cijpqklmn = Cijpqmnkl = Cklijmnpq = Cklijpqmn = Cklmnijpq = Cklmnpqij = Cklpqijmn = Cklpqmnij = Cmnijklpq = Cmnijpqkl = Cmnklijpq = Cmnklpqij = Cmnpqijkl = Cmnpqklij = Cpqijklmn = Cpqijmnkl = Cpqklijmn = Cpqklmnij = Cpqmnijkl = Cpqmnklij
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Information about the selected tensor • 2 nd rank Second-order thermo-optical effect tensor Tij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=Tij(ΔT)2 • Temperature variation ΔT and Temperature variation ΔT Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • Tij = Tji
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Information about the selected tensor • 2 nd rank Thermo-optical effect tensor tij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=tijΔT • Temperature variation ΔT Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • tij = tji
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Information about the selected tensor • 2 nd rank Verdet tensor (related to Faraday effect) Vij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=εijmVmkHk (εijm: Levi-Civita antisymmetric tensor) • Magnetic field H Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • Vij = Vji
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Information about the selected tensor • 2 nd rank Optical activity tensor gij • Axial tensor invariant under time-reversal symmetry operation • Defining equation: G=gijlilj • Direction cosines li and Direction cosines lj Optical activity coefficient G • Intrinsic symmetry symbol: e[V2] • Symmetrized indexes due to intrinsic symmetry: • gij = gji
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Information about the selected tensor • 2 nd rank Thermogyration tensor gij • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Δgij=gijT • Temperature T Optical activity tensor variation Δgij • Intrinsic symmetry symbol: e[V2] • Symmetrized indexes due to intrinsic symmetry: • gij = gji
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Information about the selected tensor • 3 rd rank Second-order dielectric susceptibility (second harmonic generation) tensor χ(2ω)ijk (without dispersion) • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi=χ(2ω)ijkEjEk • Electric field E and Electric field E Polarization vector P • Intrinsic symmetry symbol: [V3] • Symmetrized indexes due to intrinsic symmetry: • χ(2ω)ijk = χ(2ω)ikj = χ(2ω)jik = χ(2ω)jki = χ(2ω)kij = χ(2ω)kji
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Information about the selected tensor • 3 rd rank Pockels (electrooptic) effect zijk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=zijkEk • Electric field E Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2]V • Symmetrized indexes due to intrinsic symmetry: • zijk = zjik
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Information about the selected tensor • 3 rd rank Thermoelectro-optical effect tensor r(T)ijk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=r(T)ijkEkΔT • Electric field E and Temperature variation ΔT Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2]V • Symmetrized indexes due to intrinsic symmetry: • r(T)ijk = r(T)jik
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Information about the selected tensor • 3 rd rank Second-order dielectric susceptibility (second harmonic generation) tensor χ(2ω)ijk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi=χ(2ω)ijkEjEk • Electric field E and Electric field E Polarization vector P • Intrinsic symmetry symbol: V[V2] • Symmetrized indexes due to intrinsic symmetry: • χ(2ω)ijk = χ(2ω)ikj
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Information about the selected tensor • 3 rd rank Second-order dielectric susceptibility tensor (difference frequency generation) χ(0, ω, ω)ijk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi=χ(0, ω, ω)ijkEjEk • Electric field E and Electric field E Polarization vector P • Intrinsic symmetry symbol: V[V2] • Symmetrized indexes due to intrinsic symmetry: • χ(0, ω, ω)ijk = χ(0, ω, ω)ikj
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Information about the selected tensor • 3 rd rank Second-order dielectric susceptibility general tensor χijk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi=χijkEjEk • Electric field E and Electric field E Polarization vector P • Intrinsic symmetry symbol: V3
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Information about the selected tensor • 3 rd rank Magneto-optical tensor (Faraday effect) Fijk • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=FijkHk • Magnetic field H Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: e{V2}V • Symmetrized indexes due to intrinsic symmetry: • Fijk = -Fjik
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Information about the selected tensor • 3 rd rank Thermomagneto-optical effect tensor f(T)ijk • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=f(T)ijkHkΔT • Magnetic field H and Temperature variation ΔT Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: e{V2}V • Symmetrized indexes due to intrinsic symmetry: • f(T)ijk = -f(T)jik
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Information about the selected tensor • 3 rd rank Electrogyration effect tensor γijk • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Δgij=γijkEk • Electric field E Optical activity tensor variation Δgij • Intrinsic symmetry symbol: e[V2]V • Symmetrized indexes due to intrinsic symmetry: • γijk = γjik
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Information about the selected tensor • 4 th rank Third-order dielectric susceptibility (third harmonic generation) tensor χ(3ω)ijkl (without dispersion) • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi=χ(3ω)ijklEjEkEl • Electric field E and Electric field E and Electric field E Polarization vector P • Intrinsic symmetry symbol: [V4] • Symmetrized indexes due to intrinsic symmetry: • χ(3ω)ijkl = χ(3ω)ijlk = χ(3ω)ikjl = χ(3ω)iklj = χ(3ω)iljk = χ(3ω)ilkj = χ(3ω)jikl = χ(3ω)jilk = χ(3ω)jkil = χ(3ω)jkli = χ(3ω)jlik = χ(3ω)jlki = χ(3ω)kijl = χ(3ω)kilj = χ(3ω)kjil = χ(3ω)kjli = χ(3ω)klij = χ(3ω)klji = χ(3ω)lijk = χ(3ω)likj = χ(3ω)ljik = χ(3ω)ljki = χ(3ω)lkij = χ(3ω)lkji
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Information about the selected tensor • 4 th rank Elasto-optical tensor pijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=pijklεkl • Strain tensor εij Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2][V2] • Symmetrized indexes due to intrinsic symmetry: • πijkl = πjikl • πijkl = πijlk
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Information about the selected tensor • 4 th rank Kerr effect tensor Rijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=RijklEkEl • Electric field E and Electric field E Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2][V2] • Symmetrized indexes due to intrinsic symmetry: • Rijkl = Rjikl • Rijkl = Rijlk
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Information about the selected tensor • 4 th rank Piezo-optical tensor πijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=πijklσkl • Stress tensor σij Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2][V2] • Symmetrized indexes due to intrinsic symmetry: • πijkl = πjikl • πijkl = πijlk
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Information about the selected tensor • 4 th rank Second-order magneto-optical (Cotton-Mouton effect) tensor Cijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=CijklHkHl • Magnetic field H and Magnetic field H Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2][V2] • Symmetrized indexes due to intrinsic symmetry: • Cijkl = Cjikl • Cijkl = Cijlk
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Information about the selected tensor • 4 th rank Thermopiezo-optical effect tensor π(T)ijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=π(T)ijklσklΔT • Stress tensor σij and Temperature variation ΔT Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2][V2] • Symmetrized indexes due to intrinsic symmetry: • π(T)ijkl = π(T)jikl • π(T)ijkl = π(T)ijlk
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Information about the selected tensor • 4 th rank Third-order dielectric susceptibility tensor (difference frequency generation) χ(2ω, 0, ω, ω)ijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi=χ(2ω, 0, ω, ω)ijklEjEkEl • Electric field E and Electric field E and Electric field E Polarization vector P • Intrinsic symmetry symbol: [V2]V2 • Symmetrized indexes due to intrinsic symmetry: • χ(2ω, 0, ω, ω)ijkl = χ(2ω, 0, ω, ω)jikl
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Information about the selected tensor • 4 th rank Third-order dielectric susceptibility (third harmonic generation) tensor χ(3ω)ijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi=χ(3ω)ijklEjEkEl • Electric field E and Electric field E and Electric field E Polarization vector P • Intrinsic symmetry symbol: V[V3] • Symmetrized indexes due to intrinsic symmetry: • χ(3ω)ijkl = χ(3ω)ijlk = χ(3ω)ikjl = χ(3ω)iklj = χ(3ω)iljk = χ(3ω)ilkj
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Information about the selected tensor • 4 th rank Third-order dielectric susceptibility general tensor χijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi=χijklEjEkEl • Electric field E and Electric field E and Electric field E Polarization vector P • Intrinsic symmetry symbol: V4
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Information about the selected tensor • 4 th rank Magnetoelectro-optical effect tensor mijkl • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=mijklHkEl • Magnetic field H and Electric field E Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: e{V2}V2 • Symmetrized indexes due to intrinsic symmetry: • mijkl = -mjikl
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Information about the selected tensor • 4 th rank Piezogyration tensor Cijkl • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Δgij=Cijklσkl • Stress tensor σij Optical activity tensor variation Δgij • Intrinsic symmetry symbol: e[V2][V2] • Symmetrized indexes due to intrinsic symmetry: • Cijkl = Cjikl • Cijkl = Cijlk
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Information about the selected tensor • 4 th rank Quadratic electrogyration effect tensor βijkl • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Δgij=βijklEkEl • Electric field E and Electric field E Optical activity tensor variation Δgij • Intrinsic symmetry symbol: e[V2][V2] • Symmetrized indexes due to intrinsic symmetry: • βijkl = βjikl • βijkl = βijlk
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Information about the selected tensor • 5 th rank Piezoelectro-optical effect tensor zijklm • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=zijklmσklEm • Stress tensor σij and Electric field E Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2][V2]V • Symmetrized indexes due to intrinsic symmetry: • zijklm = zjiklm • zijklm = zijlkm
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Information about the selected tensor • 5 th rank Piezomagneto-optical effect tensor ωijklm • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=ωijklmHkσlm • Magnetic field H and Stress tensor σij Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: e{V2}V[V2] • Symmetrized indexes due to intrinsic symmetry: • ωijklm = -ωjiklm • ωijklm = ωijkml
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Information about the selected tensor • 5 th rank Gradient piezogyration tensor βijklm • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Δgij=βijklm∇mσkl • Stress tensor gradient ∇kσij Optical activity tensor variation Δgij • Intrinsic symmetry symbol: e[V2][V2]V • Symmetrized indexes due to intrinsic symmetry: • βijklm = βjiklm • βijklm = βijlkm
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Information about the selected tensor • 6 th rank Second-order piezo-optical tensor Πijklmn • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=Πijklmnσklσmn • Stress tensor σij and Stress tensor σij Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2][[V2][V2]] • Symmetrized indexes due to intrinsic symmetry: • Πijklmn = Πjiklmn • Πijklmn = Πijlkmn • Πijklmn = Πijklnm • Πijklmn = Πijmnkl
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Information about the selected tensor • 6 th rank Quadratic piezogyration tensor Cijklmn • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Δgij=Cijklσkl • Stress tensor σij and Stress tensor σij Optical activity tensor variation Δgij • Intrinsic symmetry symbol: e[V2][[V2][V2]] • Symmetrized indexes due to intrinsic symmetry: • Cijklmn = Cjiklmn • Cijklmn = Cijlkmn • Cijklmn = Cijklnm • Cijklmn = Cijmnkl
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Information about the selected tensor • 2 nd rank Diffusion tensor Dij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ji=Dij∇jC • Concentration gradient ∇C Diffusive flux J • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • Dij = Dji
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Information about the selected tensor • 2 nd rank Dufour effect (reversal thermodiffusion) tensor βij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: qi=βij∇jC • Concentration gradient ∇C Heat flux q • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • βij = βji
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Information about the selected tensor • 2 nd rank Electric conductivity tensor σij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ji=σijEj • Electric field E Electric current density J • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • σij = σji
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Information about the selected tensor • 2 nd rank Electric resistivity tensor ρij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ei=ρijJj • Electric current density J Electric field E • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • ρij = ρji
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Information about the selected tensor • 2 nd rank Electrodiffusion tensor γij (direct effect) • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ji=γijEjT • Electric field E Diffusive flux J • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • γij = γji
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Information about the selected tensor • 2 nd rank Electrodiffusion tensor γTij (inverse effect) • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ji=γTij∇jC • Concentration gradient ∇C Electric current density J • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • γTij = γTji
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Information about the selected tensor • 2 nd rank Soret effect (thermodiffusion) tensor βij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ji=βij∇jT • Temperature gradient ∇T Diffusive flux J • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • βij = βji
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Information about the selected tensor • 2 nd rank Thermal conductivity tensor κij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: qi=κij∇Tj • Temperature gradient ∇T Heat flux q • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • κij = κji
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Information about the selected tensor • 2 nd rank Thermal diffusivity tensor αij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: ∂T/∂t=αij∇T∇T • Temperature gradient ∇T and Temperature gradient ∇T Time derivative of the tempreature ∂T/∂t • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • αij = αji
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Information about the selected tensor • 2 nd rank Thermal resistivity tensor rij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: ∇Ti=rijqj • Heat flux q Temperature gradient ∇T • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry: • rij = rji
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Information about the selected tensor • 2 nd rank Peltier effect tensor πij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: qi=πijJj • Electric current density J Heat flux q • Intrinsic symmetry symbol: V2
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Information about the selected tensor • 2 nd rank Thermoelectric power (Seebeck effect) tensor βij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ei=βij∇Tj • Temperature gradient ∇T Electric field E • Intrinsic symmetry symbol: V2
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Information about the selected tensor • 2 nd rank Thomson heat tensor τij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: ∂q/∂t=τij∇TiJj • Temperature gradient ∇T and Electric current density J Heat production rate ∂q/∂t • Intrinsic symmetry symbol: V2
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Information about the selected tensor • 3 rd rank Hall effect (magnetorresistance) tensor Rijk • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Ei=RijkJjHk • Electric current density J and Magnetic field H Electric field E • Intrinsic symmetry symbol: e{V2}V • Symmetrized indexes due to intrinsic symmetry: • Rijk = -Rjik
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Information about the selected tensor • 3 rd rank Righi-Leduc, Maggi-Righi-Leduc and magnetothermal effects tensor Qijk • Axial tensor invariant under time-reversal symmetry operation • Defining equation: qi=Qijk∇TjHk • Temperature gradient ∇T and Magnetic field H Heat flux q • Intrinsic symmetry symbol: e{V2}V • Symmetrized indexes due to intrinsic symmetry: • Qijk = -Qjik
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Information about the selected tensor • 3 rd rank Ettinghausen effect tensor Mijk • Axial tensor invariant under time-reversal symmetry operation • Defining equation: qi=MijkJjHk • Electric current density J and Magnetic field H Heat flux q • Intrinsic symmetry symbol: eV3
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Information about the selected tensor • 3 rd rank Nernst effect tensor Nijk • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Ei=Nijk∇TjHk • Temperature gradient ∇T and Magnetic field H Electric field E • Intrinsic symmetry symbol: eV3
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Information about the selected tensor • 4 th rank Magnetic resistance tensor Tijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ei=TijklJjHkHl • Electric current density J and Magnetic field H and Magnetic field H Electric field E • Intrinsic symmetry symbol: [V2][V2] • Symmetrized indexes due to intrinsic symmetry: • Tijkl = Tjikl • Tijkl = Tijlk
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Information about the selected tensor • 4 th rank Piezoresistivity (Strain Gauge effect) tensor πijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δρij=πijklσkl • Stress tensor σij Electric resistivity tensor variation Δρij • Intrinsic symmetry symbol: [V2][V2] • Symmetrized indexes due to intrinsic symmetry: • πijkl = πjikl • πijkl = πijlk
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Information about the selected tensor • 4 th rank Magneto-Seebeck effect tensor αijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ei=αijkl∇TjHkHl • Temperature gradient ∇T and Magnetic field H and Magnetic field H Electric field E • Intrinsic symmetry symbol: V2[V2] • Symmetrized indexes due to intrinsic symmetry: • αijkl = αijlk
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