TENSOR

Information about the selected tensor
• 1 ^st rank Electric polarization vector P_i
• Polar tensor invariant under time-reversal symmetry operation
• Intrinsic symmetry symbol: V

Information about the selected tensor
• 1 ^st rank Electrocaloric effect tensor p_i
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ΔS=p_iE_i
• Electric field E Entropy variation ΔS
• Intrinsic symmetry symbol: V

Information about the selected tensor
• 1 ^st rank Heat of polarization tensor t_i
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ΔS=t_iΔP_i
• Polarization vector P variation Entropy variation ΔS
• Intrinsic symmetry symbol: V

Information about the selected tensor
• 1 ^st rank Piezoelectric polarization tensor under hydrostatic pressure d_ijj
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i=-d_ijjp
• Hydrostatic pressure p Polarization vector P
• Intrinsic symmetry symbol: V

Information about the selected tensor
• 1 ^st rank Pyroelectric tensor p_i
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ΔP_i=p_iΔT
• Temperature variation ΔT Polarization vector P variation
• Intrinsic symmetry symbol: V

Information about the selected tensor
• 2 ^nd rank Dielectric impermeability tensor β_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: E_i=β_ijD_j
• Electric displacement field D Electric field E
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• β_ij = β_ji

Information about the selected tensor
• 2 ^nd rank Dielectric permittivity tensor ε_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: D_i=ε_ijE_j
• Electric field E Electric displacement field D
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• ε_ij = ε_ji

Information about the selected tensor
• 2 ^nd rank Dielectric susceptibility tensor χ^e_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i=χ^e_ijE_j
• Electric field E Polarization vector P variation
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• χ^e_ij = χ^e_ji

Information about the selected tensor
• 2 ^nd rank Heat of deformation tensor α_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ΔS=α_ijε_ij
• Strain tensor ε_ij Entropy variation ΔS
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• α_ij = α_ji

Information about the selected tensor
• 2 ^nd rank Magnetic permeability tensor μ_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: B_i=μ^m_ijH_j
• Magnetic field H Magnetic field B
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• μ_ij = μ_ji

Information about the selected tensor
• 2 ^nd rank Magnetic susceptibility tensor χ^m_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: M_i=χ^m_ijH_j
• Magnetic field H Magnetization M
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• χ^m_ij = χ^m_ji

Information about the selected tensor
• 2 ^nd rank Piezocaloric effect tensor β_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ΔS=β_ijσ_ij
• Stress tensor σ_ij Entropy variation ΔS
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• β_ij = β_ji

Information about the selected tensor
• 2 ^nd rank Strain by hydrostatic pressure s_ijkk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ε_ij=-s_ijkkp
• Hydrostatic pressure p Strain tensor ε_ij
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• s_ij = s_ji

Information about the selected tensor
• 2 ^nd rank Strain tensor ε_ij
• Polar tensor invariant under time-reversal symmetry operation
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• ε_ij = ε_ji

Information about the selected tensor
• 2 ^nd rank Thermal expansion tensor α_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ε_ij=α_ijΔT
• Temperature variation ΔT Strain tensor ε_ij
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• α_ij = α_ji

Information about the selected tensor
• 2 ^nd rank Thermoelasticity tensor β_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: σ_ij=-β_ijΔT
• Temperature variation ΔT Stress tensor σ_ij
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• β_ij = β_ji

Information about the selected tensor
• 2 ^nd rank Toroidic susceptibility tensor τ_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: T_i=τ_ijS_j
• Toroidal field S Toroidal moment T
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• τ_ij = τ_ji

Information about the selected tensor
• 2 ^nd rank Magnetotoroidic tensor ζ_ij (direct effect)
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: M_i=ζ_ijS_j
• Toroidal field S Magnetization M
• Intrinsic symmetry symbol: eV²

Information about the selected tensor
• 2 ^nd rank Magnetotoroidic tensor ζ^T_ij (inverse effect)
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: T_i=ζ^T_ijH_j
• Magnetic field H Toroidal moment T
• Intrinsic symmetry symbol: eV²

Information about the selected tensor
• 3 ^rd rank Acoustoelectricity tensor ρ_ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: σ_ij=ρ_ijkJ_k
• Alternating electric current density J Stress tensor σ_ij
• Intrinsic symmetry symbol: [V²]V
• Symmetrized indexes due to intrinsic symmetry:
• ρ_ijk = ρ_jik
• Abbreviated notation: ρ_ijk → ρ_ij
• ij → i if i=j, ij → 9-(i+j) if i≠j

Information about the selected tensor
• 3 ^rd rank Isothermal piezoelectric tensor e^T_ijk (inverse effect)
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: σ_ij=e^T_ijkE_k
• Electric field E Stress tensor σ_ij
• Intrinsic symmetry symbol: [V²]V
• Symmetrized indexes due to intrinsic symmetry:
• e^T_ijk = e^T_jik
• Abbreviated notation: e^T_ijk → e^T_ij
• ij → i if i=j, ij → 9-(i+j) if i≠j

Information about the selected tensor
• 3 ^rd rank Piezoelectric tensor d^T_ijk (inverse effect)
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ε_ij=d^T_ijkE_k
• Electric field E Strain tensor ε_ij
• Intrinsic symmetry symbol: [V²]V
• Symmetrized indexes due to intrinsic symmetry:
• d^T_ijk = d^T_jik
• Abbreviated notation: d^T_ijk → d^T_ij
• ij → i if i=j, ij → 9-(i+j) if i≠j

Information about the selected tensor
• 3 ^rd rank Isothermal piezoelectric tensor e_ijk (direct effect)
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: D_i=e_ijkε_jk
• Strain tensor ε_ij Electric displacement field D
• Intrinsic symmetry symbol: V[V²]
• Symmetrized indexes due to intrinsic symmetry:
• e_ijk = e_ikj
• Abbreviated notation: e_ijk → e_ij
• jk → j if j=k, jk → 9-(j+k) if j≠k

Information about the selected tensor
• 3 ^rd rank Piezoelectric tensor d_ijk (direct effect)
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i=d_ijkσ_jk
• Stress tensor σ_ij Polarization vector P
• Intrinsic symmetry symbol: V[V²]
• Symmetrized indexes due to intrinsic symmetry:
• d_ijk = d_ikj
• Abbreviated notation: d_ijk → d_ij
• jk → j if j=k, jk → 9-(j+k) if j≠k

Information about the selected tensor
• 3 ^rd rank Second order magnetoelectric tensor α^T_ijk (inverse effect)
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i=α^T_ijkH_jH_k
• Magnetic field H Polarization P
• Intrinsic symmetry symbol: V[V²]
• Symmetrized indexes due to intrinsic symmetry:
• α^T_ijk = α^T_ikj

Information about the selected tensor
• 4 ^th rank Elastic compliance tensor S_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ε_ij=S_ijklσ_kl
• Stress tensor σ_ij Strain tensor ε_ij
• Intrinsic symmetry symbol: [[V²][V²]]
• Symmetrized indexes due to intrinsic symmetry:
     • S_ijkl = S_jikl
     • S_ijkl = S_ijlk
     • S_ijkl = S_klij
• Abbreviated notation: S_ijkl → S_ij
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l

Information about the selected tensor
• 4 ^th rank Elastic stiffness tensor C_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: σ_ij=C_ijklε_kl
• Strain tensor ε_ij Stress tensor σ_ij
• Intrinsic symmetry symbol: [[V²][V²]]
• Symmetrized indexes due to intrinsic symmetry:
     • C_ijkl = C_jikl
     • C_ijkl = C_ijlk
     • C_ijkl = C_klij
• Abbreviated notation: C_ijkl → C_ij
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l

Information about the selected tensor
• 4 ^th rank Viscosity tensor η_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: σ_ij=η_ijkl∂ε_kl/∂t
• Strain tensor rate ∂ε_ij/∂t Stress tensor σ_ij
• Intrinsic symmetry symbol: [[V²][V²]]
• Symmetrized indexes due to intrinsic symmetry:
     • η_ijkl = η_jikl
     • η_ijkl = η_ijlk
     • η_ijkl = η_klij
• Abbreviated notation: η_ijkl → η_ij
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l

Information about the selected tensor
• 4 ^th rank Damage effect tensor D_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: σ'_ij=D_ijklσ_kl
• Stress tensor σ_ij (before damage) Effective stress tensor σ_ij' (after damage)
• Intrinsic symmetry symbol: [V²][V²]
• Symmetrized indexes due to intrinsic symmetry:
     • D_ijkl = D_jikl
     • D_ijkl = D_ijlk
• Abbreviated notation: D_ijkl → D_ij
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l

Information about the selected tensor
• 4 ^th rank Electrostriction tensor γ_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ε_ij=γ_ijklE_kE_l
• Electric field E and Electric field E Strain tensor ε_ij
• Intrinsic symmetry symbol: [V²][V²]
• Symmetrized indexes due to intrinsic symmetry:
     • γ_ijkl = γ_jikl
     • γ_ijkl = γ_ijlk
• Abbreviated notation: γ_ijkl → γ_ij
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l

Information about the selected tensor
• 4 ^th rank Magnetostriction tensor N_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ε_ij=N_ijklM_kM_l
• Magnetization M and Magnetization M Strain tensor ε_ij
• Intrinsic symmetry symbol: [V²][V²]
• Symmetrized indexes due to intrinsic symmetry:
     • N_ijkl = N_jikl
     • N_ijkl = N_ijlk
• Abbreviated notation: N_ijkl → N_ij
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l

Information about the selected tensor
• 4 ^th rank Elastothermoelectric power tensor E_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ΔΣ_ij=E_ijklε_kl
• Strain tensor ε_ij Thermoelectric power tensor variation ΔΣ_ij
• Intrinsic symmetry symbol: V²[V²]
• Symmetrized indexes due to intrinsic symmetry:
• E_ijkl = E_ijlk
• Abbreviated notation: E_ijkl → E_ijk
• kl → k if k=l, kl → 9-(k+l) if k≠l

Information about the selected tensor
• 4 ^th rank Flexoelectric tensor μ_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i=μ_ijkl∇_lε_jk
• Strain tensor gradient ∇_kε_ij Polarization vector P
• Intrinsic symmetry symbol: V²[V²]
• Symmetrized indexes due to intrinsic symmetry:
• μ_ijkl = μ_ijlk
• Abbreviated notation: μ_ijkl → μ_ijk
• kl → k if k=l, kl → 9-(k+l) if k≠l

Information about the selected tensor
• 4 ^th rank Piezothermoelectric power tensor Π_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ΔΣ_ij=Π_ijklσ_kl
• Stress tensor σ_ij Thermoelectric power tensor variation ΔΣ_ij
• Intrinsic symmetry symbol: V²[V²]
• Symmetrized indexes due to intrinsic symmetry:
• Π_ijkl = Π_ijlk
• Abbreviated notation: Π_ijkl → Π_ijk
• kl → k if k=l, kl → 9-(k+l) if k≠l

Information about the selected tensor
• 5 ^th rank Acoustic activity tensor b_ijklm
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: σ_ij=b_ijklm∇_mε_kl
• Strain tensor gradient ∇_lε_ij Stress tensor σ_ij
• Intrinsic symmetry symbol: [[V²][V²]]V
• Symmetrized indexes due to intrinsic symmetry:
     • b_ijklm = b_jiklm
     • b_ijklm = b_ijlkm
     • b_ijklm = b_klijm
• Abbreviated notation: b_ijklm → b_ijk
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l

Information about the selected tensor
• 5 ^th rank Second-order piezoelectric tensor d_ijklm
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i=d_ijklmσ_jkσ_lm
• Stress tensor σ_ij and Stress tensor σ_ij Polarization vector P
• Intrinsic symmetry symbol: V[[V²][V²]]
• Symmetrized indexes due to intrinsic symmetry:
     • d_ijklm = d_ikjlm
     • d_ijklm = d_ijkml
     • d_ijklm = d_ilmjk
• Abbreviated notation: d_ijklm → d_ijk
     • jk → j if j=k, jk → 9-(j+k) if j≠k
     • lm → k if l=m, lm → 9-(l+m) if l≠m

Information about the selected tensor
• 6 ^th rank Third order elastic compliance tensor S_ijklmn
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ε_ij=S_ijklmnσ_klσ_mn
• Stress tensor σ_ij and Stress tensor σ_ij Strain tensor ε_ij
• Intrinsic symmetry symbol: [[V²][V²][V²]]
• Symmetrized indexes due to intrinsic symmetry:
     • S_ijklmn = S_jiklmn
     • S_ijklmn = S_ijlkmn
     • S_ijklmn = S_ijklnm
     • S_ijklmn = S_ijmnkl = S_klijmn = S_klmnij = S_mnijkl = S_mnklij
• Abbreviated notation: S_ijklmn → S_ijk
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l
     • mn → k if m=n, mn → 9-(m+n) if m≠n

Information about the selected tensor
• 6 ^th rank Third order elastic stiffness tensor C_ijklmn
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: σ_ij=C_ijklmnε_klε_mn
• Strain tensor ε_ij and Strain tensor ε_ij Stress tensor σ_ij
• Intrinsic symmetry symbol: [[V²][V²][V²]]
• Symmetrized indexes due to intrinsic symmetry:
     • C_ijklmn = C_jiklmn
     • C_ijklmn = C_ijlkmn
     • C_ijklmn = C_ijklnm
     • C_ijklmn = C_ijmnkl = C_klijmn = C_klmnij = C_mnijkl = C_mnklij
• Abbreviated notation: C_ijklmn → C_ijk
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l
     • mn → k if m=n, mn → 9-(m+n) if m≠n

Information about the selected tensor
• 8 ^th rank Damage tensor R_ijklmnpq
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: C_ijkl'=R_ijklmnpqC_mnpq
• Elastic stiffness tensor C_ijkl (before damage) Elastic stiffness tensor C_ijkl' (after damage)
• Intrinsic symmetry symbol: [[[V²][V²]][[V²][V²]]]
• Symmetrized indexes due to intrinsic symmetry:
     • R_ijklmnpq = R_jiklmnpq
     • R_ijklmnpq = R_ijlkmnpq
     • R_ijklmnpq = R_ijklnmpq
     • R_ijklmnpq = R_ijklmnqp
     • R_ijklmnpq = R_klijmnpq
     • R_ijklmnpq = R_ijklpqmn
     • R_ijklmnpq = R_mnpqijkl
• Abbreviated notation: R_ijklmnpq → R_ijkl
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l
     • mn → k if m=n, mn → 9-(m+n) if m≠n
     • pq → l if p=q, pq → 9-(p+q) if p≠q

Information about the selected tensor
• 8 ^th rank Fourth order elastic compliance tensor S_ijklmnpq
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ε_ij=S_ijklmnpqσ_klσ_mnσ_pq
• Stress tensor σ_ij and Stress tensor σ_ij and Stress tensor σ_ij Strain tensor ε_ij
• Intrinsic symmetry symbol: [[V²][V²][V²][V²]]
• Symmetrized indexes due to intrinsic symmetry:
     • S_ijklmnpq = S_jiklmnpq
     • S_ijklmnpq = S_ijlkmnpq
     • S_ijklmnpq = S_ijklnmpq
     • S_ijklmnpq = S_ijklmnqp
     • S_ijklmnpq = S_ijklpqmn = S_ijmnklpq = S_ijmnpqkl = S_ijpqklmn = S_ijpqmnkl = S_klijmnpq = S_klijpqmn = S_klmnijpq = S_klmnpqij = S_klpqijmn = S_klpqmnij = S_mnijklpq = S_mnijpqkl = S_mnklijpq = S_mnklpqij = S_mnpqijkl = S_mnpqklij = S_pqijklmn = S_pqijmnkl = S_pqklijmn = S_pqklmnij = S_pqmnijkl = S_pqmnklij
• Abbreviated notation: S_ijklmnpq → S_ijkl
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l
     • mn → k if m=n, mn → 9-(m+n) if m≠n
     • pq → l if p=q, pq → 9-(p+q) if p≠q

Information about the selected tensor
• 8 ^th rank Fourth order elastic stiffness tensor C_ijklmnpq
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: σ_ij=C_ijklmnpqε_klε_mnε_pq
• Strain tensor ε_ij and Strain tensor ε_ij and Strain tensor ε_ij Stress tensor σ_ij
• Intrinsic symmetry symbol: [[V²][V²][V²][V²]]
• Symmetrized indexes due to intrinsic symmetry:
     • C_ijklmnpq = C_jiklmnpq
     • C_ijklmnpq = C_ijlkmnpq
     • C_ijklmnpq = C_ijklnmpq
     • C_ijklmnpq = C_ijklmnqp
     • C_ijklmnpq = C_ijklpqmn = C_ijmnklpq = C_ijmnpqkl = C_ijpqklmn = C_ijpqmnkl = C_klijmnpq = C_klijpqmn = C_klmnijpq = C_klmnpqij = C_klpqijmn = C_klpqmnij = C_mnijklpq = C_mnijpqkl = C_mnklijpq = C_mnklpqij = C_mnpqijkl = C_mnpqklij = C_pqijklmn = C_pqijmnkl = C_pqklijmn = C_pqklmnij = C_pqmnijkl = C_pqmnklij
• Abbreviated notation: C_ijklmnpq → C_ijkl
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l
     • mn → k if m=n, mn → 9-(m+n) if m≠n
     • pq → l if p=q, pq → 9-(p+q) if p≠q

Information about the selected tensor
• 2 ^nd rank Second-order thermo-optical effect tensor T_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβ_ij=T_ij(ΔT)²
• Temperature variation ΔT and Temperature variation ΔT Dielectric impermeability tensor variation Δβ_ij
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• T_ij = T_ji

Information about the selected tensor
• 2 ^nd rank Thermo-optical effect tensor t_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβ_ij=t_ijΔT
• Temperature variation ΔT Dielectric impermeability tensor variation Δβ_ij
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• t_ij = t_ji

Information about the selected tensor
• 2 ^nd rank Verdet tensor (related to Faraday effect) V_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβ_ij=ε_ijmV_mkH_k (ε_ijm: Levi-Civita antisymmetric tensor)
• Magnetic field H Dielectric impermeability tensor variation Δβ_ij
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• V_ij = V_ji

Information about the selected tensor
• 2 ^nd rank Optical activity tensor g_ij
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: G=g_ijl_il_j
• Direction cosines l_i and Direction cosines l_j Optical activity coefficient G
• Intrinsic symmetry symbol: e[V²]
• Symmetrized indexes due to intrinsic symmetry:
• g_ij = g_ji

Information about the selected tensor
• 2 ^nd rank Thermogyration tensor g_ij
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Δg_ij=g_ijT
• Temperature T Optical activity tensor variation Δg_ij
• Intrinsic symmetry symbol: e[V²]
• Symmetrized indexes due to intrinsic symmetry:
• g_ij = g_ji

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: P_i=χ^(2ω)_ijkE_jE_k
• 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

Information about the selected tensor
• 3 ^rd rank Pockels (electrooptic) effect z_ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβ_ij=z_ijkE_k
• Electric field E Dielectric impermeability tensor variation Δβ_ij
• Intrinsic symmetry symbol: [V²]V
• Symmetrized indexes due to intrinsic symmetry:
• z_ijk = z_jik
• Abbreviated notation: z_ijk → z_ij
• ij → i if i=j, ij → 9-(i+j) if i≠j

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: Δβ_ij=r^(T)_ijkE_kΔT
• Electric field E and Temperature variation ΔT Dielectric impermeability tensor variation Δβ_ij
• Intrinsic symmetry symbol: [V²]V
• Symmetrized indexes due to intrinsic symmetry:
• r^(T)_ijk = r^(T)_jik
• Abbreviated notation: r^(T)_ijk → r^(T)_ij
• ij → i if i=j, ij → 9-(i+j) if i≠j

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: P_i=χ^(2ω)_ijkE_jE_k
• 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

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: P_i=χ^{(0, ω, ω)}_ijkE_jE_k
• 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

Information about the selected tensor
• 3 ^rd rank Second-order dielectric susceptibility general tensor χ_ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i=χ_ijkE_jE_k
• Electric field E and Electric field E Polarization vector P
• Intrinsic symmetry symbol: V³

Information about the selected tensor
• 3 ^rd rank Magneto-optical tensor (Faraday effect) F_ijk
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Δβ_ij=F_ijkH_k
• Magnetic field H Dielectric impermeability tensor variation Δβ_ij
• Intrinsic symmetry symbol: e{V²}V
• Symmetrized indexes due to intrinsic symmetry:
• F_ijk = -F_jik

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: Δβ_ij=f^(T)_ijkH_kΔT
• Magnetic field H and Temperature variation ΔT Dielectric impermeability tensor variation Δβ_ij
• Intrinsic symmetry symbol: e{V²}V
• Symmetrized indexes due to intrinsic symmetry:
• f^(T)_ijk = -f^(T)_jik

Information about the selected tensor
• 3 ^rd rank Electrogyration effect tensor γ_ijk
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Δg_ij=γ_ijkE_k
• Electric field E Optical activity tensor variation Δg_ij
• Intrinsic symmetry symbol: e[V²]V
• Symmetrized indexes due to intrinsic symmetry:
• γ_ijk = γ_jik

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: P_i=χ^(3ω)_ijklE_jE_kE_l
• 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

Information about the selected tensor
• 4 ^th rank Elasto-optical tensor p_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβ_ij=p_ijklε_kl
• Strain tensor ε_ij Dielectric impermeability tensor variation Δβ_ij
• Intrinsic symmetry symbol: [V²][V²]
• Symmetrized indexes due to intrinsic symmetry:
     • π_ijkl = π_jikl
     • π_ijkl = π_ijlk
• Abbreviated notation: π_ijkl → π_ij
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l

Information about the selected tensor
• 4 ^th rank Kerr effect tensor R_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβ_ij=R_ijklE_kE_l
• Electric field E and Electric field E Dielectric impermeability tensor variation Δβ_ij
• Intrinsic symmetry symbol: [V²][V²]
• Symmetrized indexes due to intrinsic symmetry:
     • R_ijkl = R_jikl
     • R_ijkl = R_ijlk
• Abbreviated notation: R_ijkl → R_ij
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l

Information about the selected tensor
• 4 ^th rank Piezo-optical tensor π_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβ_ij=π_ijklσ_kl
• Stress tensor σ_ij Dielectric impermeability tensor variation Δβ_ij
• Intrinsic symmetry symbol: [V²][V²]
• Symmetrized indexes due to intrinsic symmetry:
     • π_ijkl = π_jikl
     • π_ijkl = π_ijlk
• Abbreviated notation: π_ijkl → π_ij
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l

Information about the selected tensor
• 4 ^th rank Second-order magneto-optical (Cotton-Mouton effect) tensor C_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβ_ij=C_ijklH_kH_l
• Magnetic field H and Magnetic field H Dielectric impermeability tensor variation Δβ_ij
• Intrinsic symmetry symbol: [V²][V²]
• Symmetrized indexes due to intrinsic symmetry:
     • C_ijkl = C_jikl
     • C_ijkl = C_ijlk
• Abbreviated notation: C_ijkl → C_ij
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l

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: Δβ_ij=π^(T)_ijklσ_klΔT
• Stress tensor σ_ij and Temperature variation ΔT Dielectric impermeability tensor variation Δβ_ij
• Intrinsic symmetry symbol: [V²][V²]
• Symmetrized indexes due to intrinsic symmetry:
     • π^(T)_ijkl = π^(T)_jikl
     • π^(T)_ijkl = π^(T)_ijlk
• Abbreviated notation: π^(T)_ijkl → π^(T)_ij
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l

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: P_i=χ^{(2ω, 0, ω, ω)}_ijklE_jE_kE_l
• 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

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: P_i=χ^(3ω)_ijklE_jE_kE_l
• 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

Information about the selected tensor
• 4 ^th rank Third-order dielectric susceptibility general tensor χ_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i=χ_ijklE_jE_kE_l
• Electric field E and Electric field E and Electric field E Polarization vector P
• Intrinsic symmetry symbol: V⁴

Information about the selected tensor
• 4 ^th rank Magnetoelectro-optical effect tensor m_ijkl
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Δβ_ij=m_ijklH_kE_l
• Magnetic field H and Electric field E Dielectric impermeability tensor variation Δβ_ij
• Intrinsic symmetry symbol: e{V²}V²
• Symmetrized indexes due to intrinsic symmetry:
• m_ijkl = -m_jikl

Information about the selected tensor
• 4 ^th rank Piezogyration tensor C_ijkl
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Δg_ij=C_ijklσ_kl
• Stress tensor σ_ij Optical activity tensor variation Δg_ij
• Intrinsic symmetry symbol: e[V²][V²]
• Symmetrized indexes due to intrinsic symmetry:
     • C_ijkl = C_jikl
     • C_ijkl = C_ijlk
• Abbreviated notation: C_ijkl → C_ij
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l

Information about the selected tensor
• 4 ^th rank Quadratic electrogyration effect tensor β_ijkl
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Δg_ij=β_ijklE_kE_l
• Electric field E and Electric field E Optical activity tensor variation Δg_ij
• Intrinsic symmetry symbol: e[V²][V²]
• Symmetrized indexes due to intrinsic symmetry:
• β_ijkl = β_jikl
• β_ijkl = β_ijlk

Information about the selected tensor
• 5 ^th rank Piezoelectro-optical effect tensor z_ijklm
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβ_ij=z_ijklmσ_klE_m
• Stress tensor σ_ij and Electric field E Dielectric impermeability tensor variation Δβ_ij
• Intrinsic symmetry symbol: [V²][V²]V
• Symmetrized indexes due to intrinsic symmetry:
     • z_ijklm = z_jiklm
     • z_ijklm = z_ijlkm
• Abbreviated notation: z_ijklm → z_ijk
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l

Information about the selected tensor
• 5 ^th rank Piezomagneto-optical effect tensor ω_ijklm
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Δβ_ij=ω_ijklmH_kσ_lm
• Magnetic field H and Stress tensor σ_ij Dielectric impermeability tensor variation Δβ_ij
• Intrinsic symmetry symbol: e{V²}V[V²]
• Symmetrized indexes due to intrinsic symmetry:
     • ω_ijklm = -ω_jiklm
     • ω_ijklm = ω_ijkml
• Abbreviated notation: ω_ijklm → ω_ijkl
     • lm → l if l=m, lm → 9-(l+m) if l≠m

Information about the selected tensor
• 5 ^th rank Gradient piezogyration tensor β_ijklm
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Δg_ij=β_ijklm∇_mσ_kl
• Stress tensor gradient ∇_kσ_ij Optical activity tensor variation Δg_ij
• Intrinsic symmetry symbol: e[V²][V²]V
• Symmetrized indexes due to intrinsic symmetry:
     • β_ijklm = β_jiklm
     • β_ijklm = β_ijlkm
• Abbreviated notation: β_ijklm → β_ijk
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l

Information about the selected tensor
• 6 ^th rank Second-order piezo-optical tensor Π_ijklmn
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβ_ij=Π_ijklmnσ_klσ_mn
• Stress tensor σ_ij and Stress tensor σ_ij Dielectric impermeability tensor variation Δβ_ij
• Intrinsic symmetry symbol: [V²][[V²][V²]]
• Symmetrized indexes due to intrinsic symmetry:
     • Π_ijklmn = Π_jiklmn
     • Π_ijklmn = Π_ijlkmn
     • Π_ijklmn = Π_ijklnm
     • Π_ijklmn = Π_ijmnkl
• Abbreviated notation: Π_ijklmn → Π_ijk
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l
     • mn → k if m=n, mn → 9-(m+n) if m≠n

Information about the selected tensor
• 6 ^th rank Quadratic piezogyration tensor C_ijklmn
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Δg_ij=C_ijklσ_kl
• Stress tensor σ_ij and Stress tensor σ_ij Optical activity tensor variation Δg_ij
• Intrinsic symmetry symbol: e[V²][[V²][V²]]
• Symmetrized indexes due to intrinsic symmetry:
     • C_ijklmn = C_jiklmn
     • C_ijklmn = C_ijlkmn
     • C_ijklmn = C_ijklnm
     • C_ijklmn = C_ijmnkl
• Abbreviated notation: C_ijklmn → C_ijk
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l
     • mn → k if m=n, mn → 9-(m+n) if m≠n

Information about the selected tensor
• 2 ^nd rank Diffusion tensor D_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: J_i=D_ij∇_jC
• Concentration gradient ∇C Diffusive flux J
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• D_ij = D_ji

Information about the selected tensor
• 2 ^nd rank Dufour effect (reversal thermodiffusion) tensor β_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: q_i=β_ij∇_jC
• Concentration gradient ∇C Heat flux q
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• β_ij = β_ji

Information about the selected tensor
• 2 ^nd rank Electric conductivity tensor σ_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: J_i=σ_ijE_j
• Electric field E Electric current density J
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• σ_ij = σ_ji

Information about the selected tensor
• 2 ^nd rank Electric resistivity tensor ρ_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: E_i=ρ_ijJ_j
• Electric current density J Electric field E
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• ρ_ij = ρ_ji

Information about the selected tensor
• 2 ^nd rank Electrodiffusion tensor γ_ij (direct effect)
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: J_i=γ_ijE_jT
• Electric field E Diffusive flux J
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• γ_ij = γ_ji

Information about the selected tensor
• 2 ^nd rank Electrodiffusion tensor γ^T_ij (inverse effect)
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: J_i=γ^T_ij∇_jC
• Concentration gradient ∇C Electric current density J
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• γ^T_ij = γ^T_ji

Information about the selected tensor
• 2 ^nd rank Soret effect (thermodiffusion) tensor β_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: J_i=β_ij∇_jT
• Temperature gradient ∇T Diffusive flux J
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• β_ij = β_ji

Information about the selected tensor
• 2 ^nd rank Thermal conductivity tensor κ_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: q_i=κ_ij∇T_j
• Temperature gradient ∇T Heat flux q
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• κ_ij = κ_ji

Information about the selected tensor
• 2 ^nd rank Thermal diffusivity tensor α_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ∂T/∂t=α_ij∇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:
• α_ij = α_ji

Information about the selected tensor
• 2 ^nd rank Thermal resistivity tensor r_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ∇T_i=r_ijq_j
• Heat flux q Temperature gradient ∇T
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• r_ij = r_ji

Information about the selected tensor
• 2 ^nd rank Peltier effect tensor π_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: q_i=π_ijJ_j
• Electric current density J Heat flux q
• Intrinsic symmetry symbol: V²

Information about the selected tensor
• 2 ^nd rank Thermoelectric power (Seebeck effect) tensor β_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: E_i=β_ij∇T_j
• Temperature gradient ∇T Electric field E
• Intrinsic symmetry symbol: V²

Information about the selected tensor
• 2 ^nd rank Thomson heat tensor τ_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ∂q/∂t=τ_ij∇T_iJ_j
• Temperature gradient ∇T and Electric current density J Heat production rate ∂q/∂t
• Intrinsic symmetry symbol: V²

Information about the selected tensor
• 3 ^rd rank Hall effect (magnetorresistance) tensor R_ijk
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: E_i=R_ijkJ_jH_k
• Electric current density J and Magnetic field H Electric field E
• Intrinsic symmetry symbol: e{V²}V
• Symmetrized indexes due to intrinsic symmetry:
• R_ijk = -R_jik

Information about the selected tensor
• 3 ^rd rank Righi-Leduc, Maggi-Righi-Leduc and magnetothermal effects tensor Q_ijk
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: q_i=Q_ijk∇T_jH_k
• Temperature gradient ∇T and Magnetic field H Heat flux q
• Intrinsic symmetry symbol: e{V²}V
• Symmetrized indexes due to intrinsic symmetry:
• Q_ijk = -Q_jik

Information about the selected tensor
• 3 ^rd rank Ettinghausen effect tensor M_ijk
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: q_i=M_ijkJ_jH_k
• Electric current density J and Magnetic field H Heat flux q
• Intrinsic symmetry symbol: eV³

Information about the selected tensor
• 3 ^rd rank Nernst effect tensor N_ijk
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: E_i=N_ijk∇T_jH_k
• Temperature gradient ∇T and Magnetic field H Electric field E
• Intrinsic symmetry symbol: eV³

Information about the selected tensor
• 4 ^th rank Magnetic resistance tensor T_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: E_i=T_ijklJ_jH_kH_l
• 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:
• T_ijkl = T_jikl
• T_ijkl = T_ijlk

Information about the selected tensor
• 4 ^th rank Piezoresistivity (Strain Gauge effect) tensor π_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δρ_ij=π_ijklσ_kl
• Stress tensor σ_ij Electric resistivity tensor variation Δρ_ij
• Intrinsic symmetry symbol: [V²][V²]
• Symmetrized indexes due to intrinsic symmetry:
     • π_ijkl = π_jikl
     • π_ijkl = π_ijlk
• Abbreviated notation: π_ijkl → π_ij
     • ij → i if i=j, ij → 9-(i+j) if i≠j
     • kl → j if k=l, kl → 9-(k+l) if k≠l

Information about the selected tensor
• 4 ^th rank Magneto-Seebeck effect tensor α_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: E_i=α_ijkl∇T_jH_kH_l
• 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:
• α_ijkl = α_ijlk

TENSOR: Tensor calculation for Point Groups

For the symmetry-adapted form of crystal tensors of magnetically ordered structures see MTENSOR