MTENSOR

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
• Relates Electric field E with 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
• Relates Polarization vector P variation with 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
• Relates Hydrostatic pressure p with Polarization vector P.
• This tensor of rank 2 is obtained from the contraction of the last two indices of the elastic compliance tensor s_ijkl.
• 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
• Relates Temperature variation ΔT with Polarization vector P variation
• Intrinsic symmetry symbol: V

Information about the selected tensor
• 1 ^st rank Pyrotoroidic tensor r_i
• Polar tensor which inverts under time-reversal symmetry operation
• Defining equation: T_i=r_iΔT
• Relates Temperature variation ΔT with Toroidal moment T
• Intrinsic symmetry symbol: aV

Information about the selected tensor
• 1 ^st rank Toroidalcaloric tensor r^T_i
• Polar tensor which inverts under time-reversal symmetry operation
• Defining equation: ΔS=r^T_iS_i
• Relates Toroidal field S with Entropy variation ΔS
• Intrinsic symmetry symbol: aV

Information about the selected tensor
• 1 ^st rank Magnetocaloric tensor q^T_i
• Axial tensor which inverts under time-reversal symmetry operation
• Defining equation: ΔS=q^T_iH_i
• Relates Magnetic field H with Entropy variation ΔS
• Intrinsic symmetry symbol: aeV

Information about the selected tensor
• 1 ^st rank Pyromagnetic tensor q_i (direct effect)
• Axial tensor which inverts under time-reversal symmetry operation
• Defining equation: M_i=q_iΔT
• Relates Temperature variation ΔT with Magnetization M
• Intrinsic symmetry symbol: aeV

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
• Relates Electric displacement field D with 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
• Relates Electric field E with 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
• Relates Electric field E with 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
• Relates Strain tensor ε_ij with 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^m_ij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: B_i=μ^m_ijH_j
• Relates Magnetic field H with Magnetic field B
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• μ^m_ij = μ^m_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
• Relates Magnetic field H with 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
• Relates Stress tensor σ_ij with 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
• Relates Hydrostatic pressure p with Strain tensor ε_ij.
• This tensor of rank 2 is obtained from the contraction of the last two indices of the elastic compliance tensor s_ijkl.
• 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
• Relates Temperature variation ΔT with 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
• Relates Temperature variation ΔT with 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
• Relates Toroidal field S with 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
• Relates Toroidal field S with 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
• Relates Magnetic field H with Toroidal moment T
• Intrinsic symmetry symbol: eV²

Information about the selected tensor
• 2 ^nd rank Electrotoroidic tensor θ_ij (direct effect)
• Polar tensor which inverts under time-reversal symmetry operation
• Defining equation: P_i=θ_ijS_j
• Relates Toroidal field S with Polarization P
• Intrinsic symmetry symbol: aV²

Information about the selected tensor
• 2 ^nd rank Electrotoroidic tensor θ^T_ij (inverse effect)
• Polar tensor which inverts under time-reversal symmetry operation
• Defining equation: T_i=θ^T_ijE_j
• Relates Electric field E with Toroidal moment T
• Intrinsic symmetry symbol: aV²

Information about the selected tensor
• 2 ^nd rank Magnetoelectric tensor α_ij (direct effect)
• Axial tensor which inverts under time-reversal symmetry operation
• Defining equation: M_i=α_ijE_j
• Relates Electric field E with Magnetization M
• Intrinsic symmetry symbol: aeV²

Information about the selected tensor
• 2 ^nd rank Magnetoelectric tensor α^T_ij (inverse effect)
• Axial tensor which inverts under time-reversal symmetry operation
• Defining equation: P_i=α^T_ijH_j
• Relates Magnetic field H with Polarization P
• Intrinsic symmetry symbol: aeV²

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
• Relates Alternating electric current density J with 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
• Relates Electric field E with 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 (inversee ffect) d^T_ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ε_ij=d^T_ijkE_k
• Relates Electric field E with 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
• Relates Strain tensor ε_ij with 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(directeffect) d_ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i=d_ijkσ_jk
• Relates Stress tensor σ_ij with 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
• Relates Magnetic field H with Polarization P
• Intrinsic symmetry symbol: V[V²]
• Symmetrized indexes due to intrinsic symmetry:
• α^T_ijk = α^T_ikj

Information about the selected tensor
• 3 ^rd rank Piezotoroidic tensor γ_ijk (direct effect)
• Polar tensor which inverts under time-reversal symmetry operation
• Defining equation: ε_ij=γ_ijkS_k
• Relates Toroidal field S with Strain tensor ε_ij
• Intrinsic symmetry symbol: a[V²]V
• Symmetrized indexes due to intrinsic symmetry:
• γ_ijk = γ_jik
• Abbreviated notation: γ_ijk → γ_ij
• ij → i if i=j, ij → 9-(i+j) if i≠j

Information about the selected tensor
• 3 ^rd rank Piezotoroidic tensor γ^T_ijk (inverse effect)
• Polar tensor which inverts under time-reversal symmetry operation
• Defining equation: T_i=γ^T_ijkσ_jk
• Relates Stress tensor σ_ij with Toroidal moment T
• Intrinsic symmetry symbol: aV[V²]
• Symmetrized indexes due to intrinsic symmetry:
• γ^T_ijk = γ^T_ikj
• Abbreviated notation: γ^T_ijk → γ^T_ij
• jk → j if j=k, jk → 9-(j+k) if j≠k

Information about the selected tensor
• 3 ^rd rank Piezomagnetic tensor Λ^T_ijk (inverse effect)
• Axial tensor which inverts under time-reversal symmetry operation
• Defining equation: ε_ij=Λ^T_ijkH_k
• Relates Magnetic field H with Strain tensor ε_ij
• Intrinsic symmetry symbol: ae[V²]V
• Symmetrized indexes due to intrinsic symmetry:
• Λ^T_ijk = Λ^T_jik
• Abbreviated notation: Λ^T_ijk → Λ^T_ij
• ij → i if i=j, ij → 9-(i+j) if i≠j

Information about the selected tensor
• 3 ^rd rank Piezomagnetic tensor Λ_ijk (direct effect)
• Axial tensor which inverts under time-reversal symmetry operation
• Defining equation: M_i=Λ_ijkσ_jk
• Relates Stress tensor σ_ij with Magnetization M
• Intrinsic symmetry symbol: aeV[V²]
• Symmetrized indexes due to intrinsic symmetry:
• Λ_ijk = Λ_ikj
• Abbreviated notation: Λ_ijk → Λ_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_ijk (direct effect)
• Axial tensor which inverts under time-reversal symmetry operation
• Defining equation: M_i=α_ijkE_jE_k
• Relates Electric field E with Magnetization M
• Intrinsic symmetry symbol: aeV[V²]
• Symmetrized indexes due to intrinsic symmetry:
• α_ijk = α_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
• Relates Stress tensor σ_ij with 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 C_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: σ_ij=C_ijklε_kl
• Relates Strain tensor ε_ij with 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
• Relates Strain tensor rate ∂ε_ij/∂t with 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
• Relates Stress tensor σ_ij (before damage) with 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
• Relates Electric field E and Electric field E with 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
• Relates Magnetization M and Magnetization M with 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 Flexoelectric New μ_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i=μ_ijkl∇_jε_kl
• Relates strain tensor gradient ∇_jε_kl with polarization vector P
• According to Eliseev and Morozovska, Phys. Rev. B 98, 094108 (2018) the intrinsic symmetry of this tensor is not V2[V2] but V[V3]. This allows to further reduce the number of independent coefficients.
• Intrinsic symmetry symbol: V[V³]
• Symmetrized indexes due to intrinsic symmetry:
     • μ_ijkl = μ_ikjl
     • μ_ijkl = μ_ilkj
     • μ_ijkl = μ_ijlk
• Abbreviated notation: μ_ijkl → μ_ijk
     • jk → j if j=k, jk → 9-(j+k) if j≠k

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
• Relates Strain tensor ε_ij with 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∇_jε_kl
• Relates Strain tensor gradient ∇_iε_jk with 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
• Relates Stress tensor σ_ij with 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
• 4 ^th rank Flexomagnetic tensor Q_ijkl
• Axial tensor which inverts under time-reversal symmetry operation
• Defining equation: M_i=Q_ijkl∇_jσ_kl
• Relates Stress tensor gradient ∇_iσ_jk with Magnetization M
• Intrinsic symmetry symbol: aeV²[V²]
• Symmetrized indexes due to intrinsic symmetry:
• Q_ijkl = Q_ijlk
• Abbreviated notation: Q_ijkl → Q_ijk
• kl → k if k=l, kl → 9-(k+l) if k≠l

Information about the selected tensor
• 4 ^th rank Piezomagnetoelectric tensor π_ijkl
• Axial tensor which inverts under time-reversal symmetry operation
• Defining equation: P_i=π_ijklH_jσ_kl
• Relates Magnetization M and Stress tensor σ_ij with Polarization vector P
• Intrinsic symmetry symbol: aeV²[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
• Relates Strain tensor gradient ∇_lε_ij with 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
• Relates Stress tensor σ_ij and Stress tensor σ_ij with 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
• Relates Stress tensor σ_ij and Stress tensor σ_ij with 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
• Relates Strain tensor ε_ij and Strain tensor ε_ij with 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
• Relates Elastic stiffness tensor C_ijkl (before damage) with 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
• Relates Stress tensor σ_ij and Stress tensor σ_ij and Stress tensor σ_ij with 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
• Relates Strain tensor ε_ij and Strain tensor ε_ij and Strain tensor ε_ij with 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 Index ellipsoid β_ij
• Polar tensor invariant under time-reversal symmetry operation
• Intrinsic symmetry symbol: [V²]
• Symmetrized indexes due to intrinsic symmetry:
• β_ij = β_ji

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
• Relates Temperature variation ΔT and Temperature variation ΔT with the symmetric part of the 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
• Relates Temperature variation ΔT with the symmetric part of the 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 (with ε_ijm Levi-Civita axial antisymmetric tensor)
• Relates magnetic field H with the antisymmetric part of the dielectric impermeability tensor variation Δβ_ij
• Related with Faraday effect coefficients F: F_ijk=ε_ijmV_mk, where ε_ijm Levi-Civita axial antisymmtric tensor.
• Intrinsic symmetry symbol: V²

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
• Relates Direction cosines l_i and Direction cosines l_j with 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
• Relates Temperature T with 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
• 2 ^nd rank Spontaneous Faraday effect β_ij
• Polar tensor which inverts under time-reversal symmetry operation
• Intrinsic symmetry symbol: a{V²}
• Symmetrized indexes due to intrinsic symmetry:
• β_ij = -β_ji

Information about the selected tensor
• 3 ^rd rank Natural optical activity γ_ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβ_ij=i γ_ijkk_k
• Relates light wave vector with dielectric impermeability tensor variation (antisymmetric part).
• Connected with gyrotropic second-rank axial tensor g_lk=k₀/2ε_ijkg_ijl, with ε_ijk Levi-Civita axial antisymmetric tensor and k₀ modulus of light wave vector in vacuum. Gyration given by G=g_lkk_lk_k/k₀².
• Real in non-disipative media.
• Intrinsic symmetry symbol: {V²}V
• Symmetrized indexes due to intrinsic symmetry:
• γ_ijk = -γ_jik

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
• Relates Electric field E with the symmetric part of the 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
• Relates Electric field E and Temperature variation ΔT with the symmetric part of the 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 Magneto-optical tensor (Faraday effect) F_ijk
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Δβ_ij=F_ijkH_k
• Relates Magnetic field H with the antisymmetric part of the Dielectric impermeability tensor variation Δβ_ij.
• Pure imaginary in non-dissipative media.
• 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
• Relates Magnetic field H and Temperature variation ΔT with the antisymmetric part of the 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
• Relates Electric field E with 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
• 3 ^rd rank Spontaneous Gyrotropic Birefringence γ_ijk
• Polar tensor which inverts under time-reversal symmetry operation
• Defining equation: Δβ_ij=i γ_ijkk_k
• Relates light wave vector with dielectric impermeability tensor variation (symmetric part).
• Pure imaginary in non-dissipative media.
• Intrinsic symmetry symbol: a[V²]V
• Symmetrized indexes due to intrinsic symmetry:
• γ_ijk = γ_jik
• Abbreviated notation: γ_ijk → γ_ij
• ij → i if i=j, ij → 9-(i+j) if i≠j

Information about the selected tensor
• 3 ^rd rank Magneto-optic Kerr effect q_ijk
• Axial tensor which inverts under time-reversal symmetry operation
• Defining equation: Δβ_ij=q_ijkH_k
• Relates Magnetic field H with Dielectric impermeability tensor variation Δβ_ij (symmetric part)
• Intrinsic symmetry symbol: ae[V²]V
• Symmetrized indexes due to intrinsic symmetry:
• q_ijk = q_jik

Information about the selected tensor
• 4 ^th rank Birefringence in Cubic Crystals γ_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβ_ij= γ_ijlmk_lk_m
• Relates light wave vector and light wave vector with dielectric impermeability tensor variation (symmetric part).
• Real in non-dissipative media.
• 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 Elasto-optical tensor p_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβ_ij=p_ijklε_kl
• Relates Strain tensor ε_ij with the symmetric part of the 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
• Relates Electric field E and Electric field E with the symmetric part of the 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
• Relates Stress tensor σ_ij with the symmetric part of the 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
• Relates Magnetic field H and Magnetic field H with the symmetric part of the 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
• Relates Stress tensor σ_ij and Temperature variation ΔT with the symmetric part of the 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 Magnetoelectro-optical effect tensor m_ijkl
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Δβ_ij=m_ijklH_kE_l
• Relates Magnetic field H and Electric field E with the antisymmetric part of the 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
• Relates Stress tensor σ_ij with 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
• Relates Electric field E and Electric field E with 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
• 4 ^th rank Nonmutual Optical Activity γ_ijkl
• Polar tensor which inverts under time-reversal symmetry operation
• Defining equation: Δβ_ij= γ_ijlmk_lk_m
• Relates light wave vector and light wave vector with dielectric impermeability tensor variation (antisymmetric part).
• Pure imaginary in non-dissipative media.
• Intrinsic symmetry symbol: a{V²}[V²]
• Symmetrized indexes due to intrinsic symmetry:
     • γ_ijkl = -γ_jikl
     • γ_ijkl = γ_ijlk
• Abbreviated notation: γ_ijkl → γ_ijk
     • ij → i if i=j, ij → 9-(i+j) if i≠j

Information about the selected tensor
• 4 ^th rank Quadratic magneto-optic Kerr effect B_ijkl
• Polar tensor which inverts under time-reversal symmetry operation
• Defining equation: Δβ_ij=B_ijklH_kH_l
• Relates Magnetic field H and Magnetic field H with the antisymmetric part of the Dielectric impermeability tensor variation Δβ_ij
• Intrinsic symmetry symbol: a{V²}[V²]
• Symmetrized indexes due to intrinsic symmetry:
     • B_ijkl = -B_jikl
     • B_ijkl = B_ijlk
• Abbreviated notation: B_ijkl → B_ijk
     • ij → i if i=j, ij → 9-(i+j) if i≠j

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
• Relates Stress tensor σ_ij and Electric field E with the symmetric part of the 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
• Relates Magnetic field H and Stress tensor σ_ij with the antisymmetric part of the 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
• Relates Stress tensor gradient ∇_kσ_ij with 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
• Relates Stress tensor σ_ij and Stress tensor σ_ij with the symmetric part of 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_ijklmnσ_klσ_mn
• Relates Stress tensor σ_ij and Stress tensor σ_ij with 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
• 3 ^rd rank General second-order susceptibility (non dissipative media and no dispersion) χ(ω₃;ω₂,ω₁)_ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i(ω₃)=χ_ijk(ω₃;ω₂,ω₁)E_j(ω₂)E_k(ω₁)
• Relates Electric field of frequency ω₁ and Electric field of frequency ω₂ with Polarization of frequency ω₃.
• Tensor of real coefficients (imaginary part null). Kleinman symmetry.
• Intrinsic symmetry symbol: [V³]
• Symmetrized indexes due to intrinsic symmetry:
     • χ(ω₃;ω₂,ω₁)_ijk = χ(ω₃;ω₂,ω₁)_jik
     • χ(ω₃;ω₂,ω₁)_ijk = χ(ω₃;ω₂,ω₁)_kji
     • χ(ω₃;ω₂,ω₁)_ijk = χ(ω₃;ω₂,ω₁)_ikj
• Abbreviated notation: χ(ω₃;ω₂,ω₁)_ijk → χ(ω₃;ω₂,ω₁)_ij
     • ij → i if i=j, ij → 9-(i+j) if i≠j

Information about the selected tensor
• 3 ^rd rank Optical rectification (non-dissipative media and no dispersion) χ(0;ω,-ω)_ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i(0)=χ_ijk(0;ω,-ω)E_j(ω)E_k(-ω)
• Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 0.
• Tensor of real coefficients (imaginary part null). Kleinman symmetry.
• Intrinsic symmetry symbol: [V³]
• Symmetrized indexes due to intrinsic symmetry:
     • χ(0;ω,-ω)_ijk = χ(0;ω,-ω)_jik
     • χ(0;ω,-ω)_ijk = χ(0;ω,-ω)_kji
     • χ(0;ω,-ω)_ijk = χ(0;ω,-ω)_ikj
• Abbreviated notation: χ(0;ω,-ω)_ijk → χ(0;ω,-ω)_ij
     • ij → i if i=j, ij → 9-(i+j) if i≠j

Information about the selected tensor
• 3 ^rd rank Second-harmonic generation (non-dissipative media and no dispersion) χ(2ω;ω,ω)_ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i(2ω)=χ_ijk(2ω;ω,ω)E_j(ω)E_k(ω)
• Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω.
• Tensor of real coefficients (imaginary part null). Kleinman symmetry.
• Intrinsic symmetry symbol: [V³]
• Symmetrized indexes due to intrinsic symmetry:
     • χ(2ω;ω,ω)_ijk = χ(2ω;ω,ω)_jik
     • χ(2ω;ω,ω)_ijk = χ(2ω;ω,ω)_kji
     • χ(2ω;ω,ω)_ijk = χ(2ω;ω,ω)_ikj
• Abbreviated notation: χ(2ω;ω,ω)_ijk → χ(2ω;ω,ω)_ij
     • ij → i if i=j, ij → 9-(i+j) if i≠j

Information about the selected tensor
• 3 ^rd rank General optical rectification (dissipative media) χ(0;ω,-ω)_ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i(0)=χ_ijk(0;ω,-ω)E_j(ω)E_k(-ω)
• Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 0.
• Tensor of complex coefficients. Real in non-dissipative media.
• Onsager relations imply 1'χ_ijk(0;ω,-ω)=χ_ikj(0;ω,-ω)
• Intrinsic symmetry symbol: V[V²]*

Information about the selected tensor
• 3 ^rd rank Optical rectification (non-dissipative media)Real part χ(0;ω,-ω)_ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i(0)=χ_ijk(0;ω,-ω)E_j(ω)E_k(-ω)
• Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 0.
• Tensor of complex coefficients. Only the real part of χ_ijk(0;ω,-ω) is considered.
• Intrinsic symmetry symbol: V[V²]
• Symmetrized indexes due to intrinsic symmetry:
• χ(0;ω,-ω)_ijk = χ(0;ω,-ω)_ikj
• Abbreviated notation: χ(0;ω,-ω)_ijk → χ(0;ω,-ω)_ij
• jk → j if j=k, jk → 9-(j+k) if j≠k

Information about the selected tensor
• 3 ^rd rank Second-harmonic generation (non-dissipative media). Real part. χ(2ω;ω,ω)_ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i(2ω)=χ_ijk(2ω;ω,ω)E_j(ω)E_k(ω)
• Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω.
• Tensor of complex coefficients. Only the real part of χ_ijk(2ω;ω,ω) is considered.
• Intrinsic symmetry symbol: V[V²]
• Symmetrized indexes due to intrinsic symmetry:
• χ(2ω;ω,ω)_ijk = χ(2ω;ω,ω)_ikj
• Abbreviated notation: χ(2ω;ω,ω)_ijk → χ(2ω;ω,ω)_ij
• jk → j if j=k, jk → 9-(j+k) if j≠k

Information about the selected tensor
• 3 ^rd rank General second-order susceptibility (non-dissipative media). Real part. χ(ω₃;ω₂,ω₁)_ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i(ω₃)=χ_ijk(ω₃;ω₂,ω₁)E_j(ω₂)E_k(ω₁)
• Relates Electric field of frequency ω₁ and Electric field of frequency ω₂ with Polarization of frequency ω₃.
• Tensor of complex coefficients. Only the real part of χ_ijk(ω₃;ω₂,ω₁) is considered.
• Intrinsic symmetry symbol: V³

Information about the selected tensor
• 3 ^rd rank General second-order susceptibility by magnetic dipole (non-dissipative media and no dispersion) χ^m(ω₃;ω₂,ω₁)_ijk
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: M_i(ω₃)=χ^m_ijk(ω₃;ω₂,ω₁)E_j(ω₂)E_k(ω₁)
• Relates Electric field of frequency ω₁ and Electric field of frequency ω₂ with Magnetization of frequency ω₃.
• Tensor of pure imaginary coefficients (real part null). Kleinman symmetry.
• Intrinsic symmetry symbol: e[V³]
• Symmetrized indexes due to intrinsic symmetry:
     • χ^m(ω₃;ω₂,ω₁)_ijk = χ^m(ω₃;ω₂,ω₁)_jik
     • χ^m(ω₃;ω₂,ω₁)_ijk = χ^m(ω₃;ω₂,ω₁)_kji
     • χ^m(ω₃;ω₂,ω₁)_ijk = χ^m(ω₃;ω₂,ω₁)_ikj
• Abbreviated notation: χ^m(ω₃;ω₂,ω₁)_ijk → χ^m(ω₃;ω₂,ω₁)_ij
     • ij → i if i=j, ij → 9-(i+j) if i≠j

Information about the selected tensor
• 3 ^rd rank Second-harmonic generation by magnetic dipole (non-dissipative media and no dispersion) χ^m(2ω;ω,ω)_ijk
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: M_i(2ω)=χ^m_ijk(2ω;ω,ω)E_j(ω)E_k(ω)
• Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω.
• Tensor of pure imaginary coefficients (real part null). Kleinman symmetry.
• Intrinsic symmetry symbol: e[V³]
• Symmetrized indexes due to intrinsic symmetry:
     • χ^m(2ω;ω,ω)_ijk = χ^m(2ω;ω,ω)_jik
     • χ^m(2ω;ω,ω)_ijk = χ^m(2ω;ω,ω)_kji
     • χ^m(2ω;ω,ω)_ijk = χ^m(2ω;ω,ω)_ikj
• Abbreviated notation: χ^m(2ω;ω,ω)_ijk → χ^m(2ω;ω,ω)_ij
     • ij → i if i=j, ij → 9-(i+j) if i≠j

Information about the selected tensor
• 3 ^rd rank Second-harmonic generation by magnetic dipole (non-dissipative media). Imaginary part. χ^m(2ω;ω,ω)_ijk
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: M_i(2ω)=χ^m_ijk(2ω;ω,ω)E_j(ω)E_k(ω)
• Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω.
• Tensor of complex coefficients. Only the imaginary part of χ^m_ijk(2ω;ω,ω) is considered.
• Intrinsic symmetry symbol: eV[V²]
• Symmetrized indexes due to intrinsic symmetry:
• χ^m(2ω;ω,ω)_ijk = χ^m(2ω;ω,ω)_ikj
• Abbreviated notation: χ^m(2ω;ω,ω)_ijk → χ^m(2ω;ω,ω)_ij
• jk → j if j=k, jk → 9-(j+k) if j≠k

Information about the selected tensor
• 3 ^rd rank General second-order susceptibility by magnetic dipole (non-dissipative media) Imaginary part χ^m(ω₃;ω₂,ω₁)_ijk
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: M_i(ω₃)=χ^m_ijk(ω₃;ω₂,ω₁)E_j(ω₂)E_k(ω₁)
• Relates Electric field of frequency ω₁ and Electric field of frequency ω₂ with Magnetization of frequency ω₃.
• Tensor of complex coefficients. Only the imaginary part of χ^m_ijk(ω₃;ω₂,ω₁) is considered.
• Intrinsic symmetry symbol: eV³

Information about the selected tensor
• 3 ^rd rank Optical rectification (non-dissipative media)Imaginary part χ(0;ω,-ω)_ijk
• Polar tensor which inverts under time-reversal symmetry operation
• Defining equation: P_i(0)=χ_ijk(0;ω,-ω)E_j(ω)E_k(-ω)
• Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 0.
• Tensor of complex coefficients. Only the imaginary part of χ_ijk(0;ω,-ω) is considered.
• Intrinsic symmetry symbol: aV{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-harmonic generation (non-dissipative media). Imaginary part. χ(2ω;ω,ω)_ijk
• Polar tensor which inverts under time-reversal symmetry operation
• Defining equation: P_i(2ω)=χ_ijk(2ω;ω,ω)E_j(ω)E_k(ω)
• Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω.
• Tensor of complex coefficients. Only the imaginary part of χ_ijk(2ω;ω,ω) is considered.
• Intrinsic symmetry symbol: aV[V²]
• Symmetrized indexes due to intrinsic symmetry:
• χ(2ω;ω,ω)_ijk = χ(2ω;ω,ω)_ikj
• Abbreviated notation: χ(2ω;ω,ω)_ijk → χ(2ω;ω,ω)_ij
• jk → j if j=k, jk → 9-(j+k) if j≠k

Information about the selected tensor
• 3 ^rd rank General second-order susceptibility (non-dissipative media). Imaginary part. χ(ω₃;ω₂,ω₁)_ijk
• Polar tensor which inverts under time-reversal symmetry operation
• Defining equation: P_i(ω₃)=χ_ijk(ω₃;ω₂,ω₁)E_j(ω₂)E_k(ω₁)
• Relates Electric field of frequency ω₁ and Electric field of frequency ω₂ with Polarization of frequency ω₃.
• Tensor of complex coefficients. Only the imaginary part of χ_ijk(ω₃ω,ω) is considered.
• Intrinsic symmetry symbol: aV³

Information about the selected tensor
• 3 ^rd rank Second-harmonic generation by magnetic dipole (non-dissipative media). Real part. χ^m(2ω;ω,ω)_ijk
• Axial tensor which inverts under time-reversal symmetry operation
• Defining equation: M_i(2ω)=χ^m_ijk(2ω;ω,ω)E_j(ω)E_k(ω)
• Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω.
• Tensor of complex coefficients. Only the real part of χ^m_ijk(2ωω,ω) is considered.
• Intrinsic symmetry symbol: aeV[V²]
• Symmetrized indexes due to intrinsic symmetry:
• χ^m(2ω;ω,ω)_ijk = χ^m(2ω;ω,ω)_ikj
• Abbreviated notation: χ^m(2ω;ω,ω)_ijk → χ^m(2ω;ω,ω)_ij
• jk → j if j=k, jk → 9-(j+k) if j≠k

Information about the selected tensor
• 3 ^rd rank General second-order susceptibility by magnetic dipole (non-dissipative media) Real part χ^m(ω₃;ω₂,ω₁)_ijk
• Axial tensor which inverts under time-reversal symmetry operation
• Defining equation: M_i(ω₃)=χ^m_ijk(ω₃;ω₂,ω₁)E_j(ω₂)E_k(ω₁)
• Relates Electric field of frequency ω₁ and Electric field of frequency ω₂ with Magnetization of frequency ω₃.
• Tensor of complex coefficients. Only the real part of χ^m_ijk(ω₃;ω₂,ω₁) is considered.
• Intrinsic symmetry symbol: aeV³

Information about the selected tensor
• 3 ^rd rank General second-harmonic generation (dissipative media) χ(2ω;ω,ω)_ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i(2ω)=χ_ijk(2ω;ω,ω)E_j(ω)E_k(ω)
• Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω.
• Tensor of complex coefficients.
• Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with the tensor for difference frequency generation: 1'χ_ijk(2ω;ω,ω)=χ_jik(ω;2ω,-ω)=χ_kij(ω;2ω,-ω)
• Intrinsic symmetry symbol: (V[V²])*
• Symmetrized indexes due to intrinsic symmetry:
• χ(2ω;ω,ω)_ijk = χ(2ω;ω,ω)_ikj
• Abbreviated notation: χ(2ω;ω,ω)_ijk → χ(2ω;ω,ω)_ij
• jk → j if j=k, jk → 9-(j+k) if j≠k

Information about the selected tensor
• 3 ^rd rank General second-order susceptibility (dissipative media) χ(ω₃;ω₂,ω₁)_ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i(ω₃)=χ_ijk(ω₃;ω₂,ω₁)E_j(ω₂)E_k(ω₁)
• Relates Electric field of frequency ω₁ and Electric field of frequency ω₂ with Polarization of frequency ω₃.
• Tensor of complex coefficients.
• Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with the tensor for difference frequency generation:
1' χ_ijk(ω₃;ω₂,ω₁)=χ_jik(ω₂;ω₃,-ω₁)=χ_kij(ω₁;ω₃,-ω₂)
• Intrinsic symmetry symbol: (V³)*

Information about the selected tensor
• 4 ^th rank Degenerate four-wave mixing χ(ω;-ω,ω,ω)_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i(ω)=χ_ijkl(ω;-ω,ω,ω)E_j(-ω)E_k(ω)E_l(ω)
• Relates Electric field of frequency ω and Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency ω.
• Tensor of complex coefficients.
• Onsager relations imply: 1'χ_ijkl(ω;-ω,ω,ω)=χ_klij(ω;-ω,ω,ω).
• Intrinsic symmetry symbol: [[V²][V²]]*
• Symmetrized indexes due to intrinsic symmetry:
     • χ(ω;-ω,ω,ω)_ijkl = χ(ω;-ω,ω,ω)_jikl
     • χ(ω;-ω,ω,ω)_ijkl = χ(ω;-ω,ω,ω)_ijlk
• Abbreviated notation: χ(ω;-ω,ω,ω)_ijkl → χ(ω;-ω,ω,ω)_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 Electric-field induced second-harmonic generation (non-dissipative media and no dispersion) χ(2ω;0,ω,ω)_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i(2ω)=χ_ijkl(2ω;0,ω,ω)E_j(0)E_k(ω)E_l(ω)
• Relates Electric field of frequency 0 and Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω.
• Tensor of real coefficients (imaginary part null). Kleinman symmetry.
• Intrinsic symmetry symbol: [V⁴]
• Symmetrized indexes due to intrinsic symmetry:
     • χ(2ω;0,ω,ω)_ijkl = χ(2ω;0,ω,ω)_jikl
     • χ(2ω;0,ω,ω)_ijkl = χ(2ω;0,ω,ω)_kjil
     • χ(2ω;0,ω,ω)_ijkl = χ(2ω;0,ω,ω)_ljki
     • χ(2ω;0,ω,ω)_ijkl = χ(2ω;0,ω,ω)_ikjl
     • χ(2ω;0,ω,ω)_ijkl = χ(2ω;0,ω,ω)_ilkj
     • χ(2ω;0,ω,ω)_ijkl = χ(2ω;0,ω,ω)_ijlk
• Abbreviated notation: χ(2ω;0,ω,ω)_ijkl → χ(2ω;0,ω,ω)_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 General third-order susceptibility (non-dissipative media and no dispersion) χ(ω₄;ω₃,ω₂,ω₁)_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i(ω₄)=χ_ijkl(ω₄;ω₃,ω₂,ω₁)E_j(ω₃)E_k(ω₂)E_l(ω₁)
• Relates Electric field of frequency ω₁ and Electric field of frequency ω₂ and Electric field of frequency ω₃ with Polarization of frequency ω₄.
• Tensor of real coefficients (imaginary part null). Kleinman symmetry.
• Intrinsic symmetry symbol: [V⁴]
• Symmetrized indexes due to intrinsic symmetry:
     • χ(ω₄;ω₃,ω₂,ω₁)_ijkl = χ(ω₄;ω₃,ω₂,ω₁)_jikl
     • χ(ω₄;ω₃,ω₂,ω₁)_ijkl = χ(ω₄;ω₃,ω₂,ω₁)_kjil
     • χ(ω₄;ω₃,ω₂,ω₁)_ijkl = χ(ω₄;ω₃,ω₂,ω₁)_ljki
     • χ(ω₄;ω₃,ω₂,ω₁)_ijkl = χ(ω₄;ω₃,ω₂,ω₁)_ikjl
     • χ(ω₄;ω₃,ω₂,ω₁)_ijkl = χ(ω₄;ω₃,ω₂,ω₁)_ilkj
     • χ(ω₄;ω₃,ω₂,ω₁)_ijkl = χ(ω₄;ω₃,ω₂,ω₁)_ijlk
• Abbreviated notation: χ(ω₄;ω₃,ω₂,ω₁)_ijkl → χ(ω₄;ω₃,ω₂,ω₁)_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-harmonic generation (non-dissipative media and no dispersion) χ(3ω;ω,ω,ω)_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i(3ω)=χ_ijkl(3ω;ω,ω,ω)E_j(ω)E_k(ω)E_l(ω)
• Relates Electric field of frequency ω and Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 3ω.
• Tensor of real coefficients (imaginary part null). Kleinman symmetry.
• Intrinsic symmetry symbol: [V⁴]
• Symmetrized indexes due to intrinsic symmetry:
     • χ(3ω;ω,ω,ω)_ijkl = χ(3ω;ω,ω,ω)_jikl
     • χ(3ω;ω,ω,ω)_ijkl = χ(3ω;ω,ω,ω)_kjil
     • χ(3ω;ω,ω,ω)_ijkl = χ(3ω;ω,ω,ω)_ljki
     • χ(3ω;ω,ω,ω)_ijkl = χ(3ω;ω,ω,ω)_ikjl
     • χ(3ω;ω,ω,ω)_ijkl = χ(3ω;ω,ω,ω)_ilkj
     • χ(3ω;ω,ω,ω)_ijkl = χ(3ω;ω,ω,ω)_ijlk
• Abbreviated notation: χ(3ω;ω,ω,ω)_ijkl → χ(3ω;ω,ω,ω)_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 Four-wave mixing χ(ω₁;-ω₂,ω₁,ω₂)_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i(ω₁)=χ_ijkl(ω₁;-ω₂,ω₁,ω₂)E_j(-ω₂)E_k(ω₁)E_l(ω₂)
• Relates Electric field of frequency ω₁ and Electric field of frequency ω₂ and Electric field of frequency ω₂ with Polarization of frequency ω₁.
• Tensor of complex coefficients.
• Onsager relations imply: 1'χ_ijkl(ω₁;-ω₂,ω₁,ω₂)=χ_klij(ω₁;-ω₂,ω₁,ω₂).
• Intrinsic symmetry symbol: [V²V²]*

Information about the selected tensor
• 4 ^th rank Third-harmonic generation (non-dissipative media). Real part. χ(3ω;ω,ω,ω)_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i(3ω)=χ_ijkl(3ω;ω,ω,ω)E_j(ω)E_k(ω)E_l(ω)
• Relates Electric field of frequency ω and Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 3ω.
• Tensor of complex coefficients. Only the real part of χ_ijkl(3ω;ω,ω,ω) is considered.
• Intrinsic symmetry symbol: V[V³]
• Symmetrized indexes due to intrinsic symmetry:
     • χ(3ω;ω,ω,ω)_ijkl = χ(3ω;ω,ω,ω)_ikjl
     • χ(3ω;ω,ω,ω)_ijkl = χ(3ω;ω,ω,ω)_ilkj
     • χ(3ω;ω,ω,ω)_ijkl = χ(3ω;ω,ω,ω)_ijlk
• Abbreviated notation: χ(3ω;ω,ω,ω)_ijkl → χ(3ω;ω,ω,ω)_ijk
     • jk → j if j=k, jk → 9-(j+k) if j≠k

Information about the selected tensor
• 4 ^th rank General third-order susceptibility (non-dissipative media). Real part. χ(ω₄;ω₃,ω₂,ω₁)_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i(ω₄)=χ_ijkl(ω₄;ω₃,ω₂,ω₁)E_j(ω₃)E_k(ω₂)E_l(ω₁)
• Relates Electric field of frequency ω₁ and Electric field of frequency ω₂ and Electric field of frequency ω₃ with Polarization of frequency ω₄.
• Tensor of complex coefficients.
• Only the real part of χ_ijkl(ω₄;ω₃,ω₂,ω₁) is considered.
• Intrinsic symmetry symbol: V⁴

Information about the selected tensor
• 4 ^th rank Electric-field induced second-harmonic generation (non-dissipative media). Real part. χ(2ω;0,ω,ω)_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i(2ω)=χ_ijkl(2ω;0,ω,ω)E_j(0)E_k(ω)E_l(ω)
• Relates Electric field of frequency 0 and Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω.
• Tensor of complex coefficients. Only the real part of χ_ijkl(2ω0,ω,ω) is considered.
• Intrinsic symmetry symbol: V²[V²]
• Symmetrized indexes due to intrinsic symmetry:
• χ(2ω;0,ω,ω)_ijkl = χ(2ω;0,ω,ω)_ijlk
• Abbreviated notation: χ(2ω;0,ω,ω)_ijkl → χ(2ω;0,ω,ω)_ijk
• kl → k if k=l, kl → 9-(k+l) if k≠l

Information about the selected tensor
• 4 ^th rank Third-harmonic generation (non-dissipative media). Imaginary part. χ(3ω;ω,ω,ω)_ijkl
• Polar tensor which inverts under time-reversal symmetry operation
• Defining equation: P_i(3ω)=χ_ijkl(3ω;ω,ω,ω)E_j(ω)E_k(ω)E_l(ω)
• Relates Electric field of frequency ω and Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 3ω.
• Tensor of complex coefficients. Only the imaginary part of χ_ijkl(3ω;ω,ω,ω) is considered.
• Intrinsic symmetry symbol: aV[V³]
• Symmetrized indexes due to intrinsic symmetry:
     • χ(3ω;ω,ω,ω)_ijkl = χ(3ω;ω,ω,ω)_ikjl
     • χ(3ω;ω,ω,ω)_ijkl = χ(3ω;ω,ω,ω)_ilkj
     • χ(3ω;ω,ω,ω)_ijkl = χ(3ω;ω,ω,ω)_ijlk
• Abbreviated notation: χ(3ω;ω,ω,ω)_ijkl → χ(3ω;ω,ω,ω)_ijk
     • jk → j if j=k, jk → 9-(j+k) if j≠k

Information about the selected tensor
• 4 ^th rank General third-order susceptibility (non-dissipative media). Imaginary part. χ(ω₄;ω₃,ω₂,ω₁)_ijkl
• Polar tensor which inverts under time-reversal symmetry operation
• Defining equation: P_i(ω₄)=χ_ijkl(ω₄;ω₃,ω₂,ω₁)E_j(ω₃)E_k(ω₂)E_l(ω₁)
• Relates Electric field of frequency ω₁ and Electric field of frequency ω₂ and Electric field of frequency ω₃ with Polarization of frequency ω₄.
• Tensor of complex coefficients.
• Only the imaginary part of χ_ijkl(ω₄;ω₃,ω₂,ω₁)is considered.
• Intrinsic symmetry symbol: aV⁴

Information about the selected tensor
• 4 ^th rank Electric-field induced second-harmonic generation (non-dissipative media). Imaginary part. χ(2ω0,ω,ω)_ijkl
• Polar tensor which inverts under time-reversal symmetry operation
• Defining equation: P_i(2ω)=χ_ijkl(2ω;0,ω,ω)E_j(0)E_k(ω)E_l(ω)
• Relates Electric field of frequency 0 and Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω.
• Tensor of complex coefficients.Only the imaginary part of χ_ijkl(2ω;0,ω,ω) is considered.
• Intrinsic symmetry symbol: aV²[V²]
• Symmetrized indexes due to intrinsic symmetry:
• χ(2ω;0,ω,ω)_ijkl = χ(2ω;0,ω,ω)_ijlk
• Abbreviated notation: χ(2ω;0,ω,ω)_ijkl → χ(2ω;0,ω,ω)_ijk
• kl → k if k=l, kl → 9-(k+l) if k≠l

Information about the selected tensor
• 4 ^th rank General third-harmonic generation (dissipative media) χ(3ω;ω,ω,ω)_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i(3ω)=χ_ijkl(3ω;ω,ω,ω)E_j(ω)E_k(ω)E_l(ω)
• Relates Electric field of frequency ω and Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 3ω.
• Tensor of complex coefficients.
• Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with a tensor for a different third order process: 1'χ_ijkl(3ω;ω,ω,ω)=χ_jikl(ω;3ω,-ω,-ω)=χ_kijl(ω;3ω,-ω,-ω)=χ_lijk(ω;3ω,-ω,-ω).
• Intrinsic symmetry symbol: (V[V³])*
• Symmetrized indexes due to intrinsic symmetry:
     • χ(3ω;ω,ω,ω)_ijkl = χ(3ω;ω,ω,ω)_ikjl
     • χ(3ω;ω,ω,ω)_ijkl = χ(3ω;ω,ω,ω)_ilkj
     • χ(3ω;ω,ω,ω)_ijkl = χ(3ω;ω,ω,ω)_ijlk
• Abbreviated notation: χ(3ω;ω,ω,ω)_ijkl → χ(3ω;ω,ω,ω)_ijk
     • jk → j if j=k, jk → 9-(j+k) if j≠k

Information about the selected tensor
• 4 ^th rank General third-order susceptibility (dissipative media) χ(ω₄;ω₃,ω₂,ω₁)_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: P_i(ω₄)=χ_ijkl(ω₄;ω₃,ω₂,ω₁)E_j(ω₃)E_k(ω₂)E_l(ω₁)
• Relates Electric field of frequency ω₁ and Electric field of frequency ω₂ and Electric field of frequency ω₃ with Polarization of frequency ω₄.
• Tensor of complex coefficients.
• Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with a tensor for other third order processes: 1'χ_ijkl(ω₄;ω₃,ω₂,ω₁)=χ_jikl(ω₃;ω₄,-ω₂,-ω₁)=χ_kijl(ω₂;ω₄,-ω₃,-ω₁)=χ_lijk(ω₁;ω₄,-ω₃,-ω₂).
• Intrinsic symmetry symbol: (V⁴)*

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
• Relates Concentration gradient ∇C with Diffusive flux J
• Intrinsic symmetry symbol: [V²]*

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
• Relates Concentration gradient ∇C with Heat flux q
• Intrinsic symmetry symbol: [V²]*

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
• Relates Electric field E with Electric current density J
• Intrinsic symmetry symbol: [V²]*

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
• Relates Electric current density J with Electric field E
• Intrinsic symmetry symbol: [V²]*

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
• Relates Electric field E with Diffusive flux J
• Intrinsic symmetry symbol: [V²]*

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
• Relates Concentration gradient ∇C with Electric current density J
• Intrinsic symmetry symbol: [V²]*

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
• Relates Temperature gradient ∇T with Diffusive flux J
• Intrinsic symmetry symbol: [V²]*

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
• Relates Temperature gradient ∇T with Heat flux q
• Intrinsic symmetry symbol: [V²]*

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
• Relates Temperature gradient ∇T and Temperature gradient ∇T with Time derivative of the temperature ∂T/∂t
• Intrinsic symmetry symbol: [V²]*

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
• Relates Heat flux q with Temperature gradient ∇T
• Intrinsic symmetry symbol: [V²]*

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
• Relates Electric current density J with Heat flux q.
• Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with the Thermoelectric power (Seebeck) effect tensor β.
•1'π_ijkl=β_jikl.
• 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
• Relates Electric field E with Temperature gradient ∇T.
• Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with the Peltier effect tensor π.
•1'β_ijkl=π_jikl
• 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
• Relates Temperature gradient ∇T and Electric current density J with Heat production rate ∂q/∂t.
• Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with another tensor property.
• 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
• Relates Electric current density J and Magnetic field H with Electric field E
• Intrinsic symmetry symbol: e{V²}*V

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
• Relates Temperature gradient ∇T and Magnetic field H with Heat flux q
• Intrinsic symmetry symbol: e{V²}*V

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
• Relates Electric current density J and Magnetic field H with Heat flux q.
• Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with the Nernst effect tensor N.
•1'M_ijkl=N_jikl.
• 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
• Relates Temperature gradient ∇T and Magnetic field H with Electric field E.
• Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with the Ettinghausen effect tensor M.
•1'N_ijkl=M_jikl.
• 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
• Relates Electric current density J and Magnetic field H and Magnetic field H with Electric field E
• Intrinsic symmetry symbol: [V²]*[V²]
• Symmetrized indexes due to intrinsic symmetry:
• T_ijkl = T_ijlk

Information about the selected tensor
• 4 ^th rank Magneto-heat-conductivity tensor S_ijkl S_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: q_i=S_ijkl(∇T)_jH_kH_l
• Relates Temperature gradient (∇T), Magnetic field H and Magnetic field H with Heat flux q.
• Onsager relations imply 1'S_ij=S_ji
• Intrinsic symmetry symbol: [V²]*[V²]
• Symmetrized indexes due to intrinsic symmetry:
• S_ijkl = S_ijlk
• Abbreviated notation: S_ijkl → S_ijk
• kl → k if k=l, kl → 9-(k+l) if k≠l

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
• Relates Stress tensor σ_ij with Electric resistivity tensor variation Δρ_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
• 4 ^th rank Magneto Peltier effect P_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: q_i=P_ijklH_kH_lJ_j
• Relates current density and magnetic field and magnetic field with heat flux.
• Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with the magneto-Seebeck effect tensor α.
•1'P_ijkl=α_jikl
• Intrinsic symmetry symbol: (V²[V²])*
• Symmetrized indexes due to intrinsic symmetry:
• P_ijkl = P_ijlk
• Abbreviated notation: P_ijkl → P_ijk
• kl → k if k=l, kl → 9-(k+l) if k≠l

Information about the selected tensor
• 4 ^th rank Magneto Seebeck effect α_ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: E_i=α_ijkl(∇T)_jH_kH_l
• Relates temperature gradient ∇T and magnetic field and magnetic field with electric field.
• Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with the magneto-Peltier effect tensor P.
•1'α_ijkl=P_jikl
• Intrinsic symmetry symbol: (V²[V²])*
• Symmetrized indexes due to intrinsic symmetry:
• α_ijkl = α_ijlk
• Abbreviated notation: α_ijkl → α_ijk
• kl → k if k=l, kl → 9-(k+l) if k≠l

MTENSOR: Tensor calculation for Magnetic Point Groups

For the symmetry-adapted form of non-magnetic crystal tensors see TENSOR