Tables for
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, List of terms and symbols.

List of terms and symbols used in this volume

A. Authiera*

Institut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France
Correspondence e-mail:

(1) Vector spaces and tensor analysis

Basis vectors in direct space (covariant) [{\bf e}_i], [{\bf a}_i]
Basis vectors in reciprocal space (contravariant) [{\bf e}^i], [{\bf a}_i^*]
Contravariant components of vectors in direct space [ x^i]
Covariant components of vectors in reciprocal space [ x_i]
Direction indices (of a lattice row) [[uvw]]
Dual (or reciprocal) space ([n] dimensions) [E_n] (Chapter 1.1[link] )
Element of [\in]
Euclidian space, direct space ([n] dimensions) [E^n]
Hermitian conjugate of matrix [M] [M^+]
Integers (positive) [\bb Z^+]
Integers (ring of) [\bb Z]
Kronecker symbol [\delta_i^j]
Metric tensor [g_{ij}]
Miller indices (of a lattice plane) [(hkl)]
Nabla operator [\nabla]
Orthogonal transformation [R]
Outer product [\bigwedge]
Partial derivative with respect to [x_i] [\partial_i]
Permutation tensor [\varepsilon_{ijk}], [{\hat e}_{ijk}]
Position vector in reciprocal space [{\bf G}], [{\bf k}]
Reciprocal lattice vector [{\bf g}_{hkl}]
Sum of spaces [\oplus]
Tensor of rank [n], [p] times covariant and [q] times contravariant ([n = p+q]) [t_{i_1 \ldots\, i_p}^{\,j_1 \ldots\, j_q}]
Tensor product [\otimes]
Transpose of matrix [M] [M^T]
Unit transformation, matrix or element [E]
Vector in superspace [{\bf a}_{\rm si}]
Vector in reciprocal superspace [{\bf a}_{\rm si}^*]
Vector product [\wedge], [\times]
Volume element d[\tau]
Volume of unit cell in direct (reciprocal) space [V] ([V^{\ast}])

(2) Group theory

Character [\chi]
Character (irreducible) [\chi_{\alpha}]
Character (value at [R]) [\chi\, (R)]
Class multiplication constants [c_{ijk}]
Conjugacy class [C_i]
Cyclic group of order [m] [C_m]
Dihedral group of order 2[n] [D_n]
Dimension of irreducible representation [\alpha] [d_{\alpha}]
Lattice translation subgroup [T(n)]
Matrix representation of point group [K] [\Gamma (K)]
Multiplicity [m_{\alpha}]
Octahedral group [O]
Order of class [C_i] [n_i]
Orthogonal group [O(n)]
Orthogonal group (special) [SO(n)]
Physically irreducible representation [R]-irep
Point group [K] (Chapter 1.2[link] ), [G_o] (Chapter 2.1[link] ), [G] (Part 3[link] )
Point group (order of) [|K|], [N]
Representation of point group [K] [D(K)]
Space group [G], [\cal G] (Part 3[link] )
Tetrahedral group [T]

(3) Physical properties

(a) Elastic properties

Bulk modulus (volume isothermal compressibility) [\kappa]
Components of the displacement vector [u_i]
Elastic compliances (second-order) [s_{ijkl}]
Elastic compliances (second-order adiabatic) [(s_{ijkl})^{\sigma}]
Elastic compliances (second-order reduced) [s_{\alpha\beta}]
Elastic compliances (third-order) [s_{ijkl mn}]
Elastic stiffnesses (second-order) [c_{ijkl}], [C_{ijkl}]
Elastic stiffnesses (second-order adiabatic) [(c_{ijkl})^{\sigma}]
Elastic stiffnesses (second-order reduced) [c_{\alpha\beta}]
Elastic stiffnesses (third-order) [c_{ijkl mn}]
Lamé coefficients [\lambda]
Normal stress [{\vec \nu}]
Poisson's ratio [\nu]
Pressure [p]
Shear stress [{\vec \tau}]
Strain tensor [S_{ij}], [u_{ij}] (Chapters 1.4[link] , 1.5[link] and 3.1[link] ), [\eta_{ij}] (Chapter 2.3[link] )
Strain Voigt matrix [S_{\alpha}]
Stress tensor [T_{ij}], [\tau_{ij}] (Chapter 1.4[link] ), [\sigma_{ij}] (Chapters 2.1[link] , 2.3[link] , 2.4[link] )
Stress Voigt matrix [T_{\alpha}]
Velocity of sound [v]
Volume [V]
Volumic mass [\rho]
Young's modulus [E]

(b) Electric properties

Charge density [\rho({\bf r})]
Charge of the electron [e]
Current density [{\bf j}({\bf r})], [J]
Dielectric impermeability [\eta_{ij}]
Dielectric permittivity or constant [\varepsilon]
Dielectric permittivity of vacuum [\varepsilon_0]
Dielectric permittivity tensor [\varepsilon_{ij}]
Dielectric permittivity tensor (adiabatic) [(\varepsilon_{ij})^\sigma ]
Dielectric susceptibility [\chi^e_{ij}], [\chi_{ijk\cdots}]
Dielectric susceptibility (nth-order) [\chi^{(n)}]
Effective mass of the electron [m^*]
Electric dipole operator [{\hat p}]
Electric displacement [{\bf D}]
Electric field [{\bf E}]
Electric polarization [{\bf P}]
Electric polarization (nth-order) [{\bf P}_{n}]
Electric polarization (nonlinear) [{\bf P}^{\rm NL}]
Electro-optic tensor [r_{ijk}]
Electrostriction tensor [Q_{ijkl}]
Electrostriction tensor (reduced) [Q_{\alpha\beta}]
Hall constant [R_{H\ \ ijk}]
Piezoelectric tensor [d_{ijk}]
Piezoelectric tensor at constant strain [e_{ijk}]
Piezoelectric tensor (reduced) [d_{i\alpha}]
Piezoelectric tensor (reduced adiabatic) [(d_{ijk})^{\sigma}]
Piezoelectric tensor (reduced inverse) [d_{\alpha i}]
Pyroelectric tensor [p_{i}]

(c) Magnetic properties

Antiferromagnetic vector [{\bf L}_i]
Bohr magneton [\mu_B]
Constant describing magnetostriction [\lambda]
Effective number of Bohr magnetons [p] (Section 1.6.1[link] )
Landé [g]-factor [g]
Magnetic birefringence [\Delta n]
Magnetic field [{\bf H}]
Magnetic induction [{\bf B}]
Magnetic moment [\boldmu]
Magnetic moment density [{\bf m}({\bf r})]
Magnetic permeability [\mu_{ij}]
Magnetic permeability of vacuum [\mu_{o}]
Magnetic susceptibility [\chi_{ij}], [\chi^{\,m}_{ij}]
Magnetization (= magnetic moment per unit volume = ferromagnetic vector) [{\bf M}]
Magnetoelastic energy [U_{\rm me}]
Magnetoelectric tensor (linear) [\alpha_{ij}]
Magnetoelectric tensor (nonlinear) [EHH] [\beta_{ijk}]
Magnetoelectric tensor (nonlinear) [HEE] [\gamma_{ijk}]
Magneto-optic tensor [\bf f]
Néel temperature [T_N]
Orbital angular momentum [{\bf L}] (Section[link] )
Piezomagnetic components [\Lambda_{ijk}]
Piezomagnetic components (reduced) [\Lambda_{i\alpha}]
Piezomagnetoelectric tensor [\pi_{ijkl}]
Spin angular momentum (of an atom or ion) [{\bf S}]
Spin density [{\bf S}({\bf r})]
Sum of the magnetic moments in a unit cell [{\bf m}]
Sum of the magnetic moments in a unit cell, in which some of the moments are taken with opposite sign [{\bf l}_i]
Total angular momentum [{\bf J}]
Weiss constant [\Delta]

(d) Optical properties

Angle between optic axes [2V]
Cyclic (or circular) frequency [\omega]
Elasto-optic (strain-optic) tensor [p_{ijkl}]
Elasto-optic (strain-optic) tensor, reduced [p_{\alpha\beta}]
Electro-optic tensor [r_{ijk}]
Ellipticity of wave [\kappa]
Gyration susceptibility [\gamma_{ijl}]
Gyration tensor [g_{ij}], [G_{ij}]
Gyration vector G
Optical rotatory power [\rho]
Phase difference of light [\Delta]
Piezo-optic tensor [\pi_{ijkl}]
Piezo-optic tensor (reduced) [\pi_{\alpha\beta}]
Polarizability operator [{\hat \alpha}]
Poynting vector S
Poynting vector (unit) s, [{\hat {\bf s}}]
Raman tensor [R^{\,j}({\bf q})]
Rayleigh length [Z_r]
Refractive index (extraordinary) [n_e]
Refractive index of light [n]
Refractive index (ordinary) [n_o]
Refractive indices for biaxial indicatrix [n_x, n_\alpha, \alpha]; [n_y, n_\beta, \beta]; [n_z, n_\gamma, \gamma]
Velocity of light in a vacuum [c]
Velocity (group) [v_g]
Wavelength of light [\lambda]
Wavevector of light propagating in crystal k ([|k| = 2\pi/\lambda])

(e) Thermodynamic properties

Anisotropy energy [U_a]
Atomic Debye–Waller factor (static) [S_\alpha ]
Atomic Debye–Waller factor (thermal) [T_\alpha ]
Boltzmann constant [k_B]
Debye frequency [\omega_D]
Debye temperature [\Theta_D]
Einstein frequency [\omega_E]
Einstein temperature [\Theta_E]
Elastic energy [U_{\rm el}]
Entropy [\sigma], [S]
Free energy [{\cal G}], [{\cal F}], [F], [A]
Grüneisen parameter [{\bar \gamma}], [{\gamma}]
Grüneisen parameter (averaged mode) [\gamma_{{\bf q}, j}]
Grüneisen parameter (generalized mode) [\gamma_{{\bf q}j,kl}]
Hamiltonian [H]
Heat current [J_Q]
Internal energy [U], [{\cal U}]
Lattice energy [E_{\rm ph}]
Partition function [Z]
Phonon wavevector q
Seebeck coefficient [S]
Specific heat at constant strain (volume) [c^{\,S}], [c_V]
Specific heat at constant stress (pressure) [c^{T}], [c_{p}]
Specific heat at constant volume (according to the Debye model) [c_V^{\rm Debye} ]
Specific heat at constant volume (according to the Einstein model) [c_V^{\rm Einstein} ]
Temperature [\Theta], [T]
Temperature-stress components [\lambda _{ij}]
Thermal conductivity [K]
Thermal expansion [\alpha_{ij}]
Thermal expansion (volume) [\beta]
Thermodynamic potential [\Phi]
Zero-point energy [E_o]

(f) Local crystal susceptibilities

Local susceptibility tensor in direct space [\chi({\bf r})]
Fourier components of the local susceptibility tensor [\chi({\bf H})]
Dipole–dipole tensor atomic factor [D_{jk}]
Symmetric part of the dipole–dipole tensor atomic factor [D_{jk}^+]
Antisymmetric part of the dipole–dipole tensor atomic factor [D_{jk}^-]
Third-rank tensor describing the dipole–quadrupole resonant X-ray scattering [I_{jkl}]
Part of the third-rank tensor invariant under time inversion and symmetric under the permutation of j and k [I_{jkl}^{++}]
Part of the third-rank tensor non-invariant under time inversion and symmetric under the permutation of j and k [I_{jkl}^{+-}]
Part of the third-rank tensor invariant under time inversion and antisymmetric under the permutation of j and k [I_{jkl}^{-+}]
Part of the third-rank tensor non-invariant under time inversion and antisymmetric under the permutation of j and k [I_{jkl}^{--}]
Fourth-rank tensor describing the quadrupole–quadrupole resonant X-ray scattering [Q_{ijkl}]

(4) Phase transformations and twinning

Order parameter (primary) [\eta]
Order parameter (secondary) [\lambda]
Point group of ferroic (low-symmetry) phase [F] (Chapters 3.1[link] and 3.4[link] )
Point group of parent (high-symmetry) phase [G]
Space group of ferroic (low-symmetry) phase [{\cal F}] (Chapters 3.1[link] and 3.4[link] )
Space group of parent (high-symmetry) phase [{\cal G}]
Symmetry descent from [G] to [F] (point groups) [G \Downarrow F]
Symmetry descent from [\cal G] to [{\cal F}] (space groups) [{\cal G} \Downarrow {\cal F}]
Eigensymmetry of untwinned crystal or daughter phase [{\cal H}] (Chapter 3.3[link] )
Transition temperature, in particular: Curie temperature [T_C]
Order of group [{\cal G}], [{\cal H}], [{\cal K}] [\vert {\cal G}\vert], [\vert {\cal H}\vert], [\vert {\cal K}\vert]
Index of [{\cal H}] in [{\cal G}], or of [{\cal H}] in [{\cal K}] [i]
Aizu symbol of a ferroic phase transition (ferroic species); F = ferroic [{\cal G}F{\cal H}] or [{\cal G}\,\gt\,F]
Composite symmetry group of a twinned crystal (domain pair); twin symmetry [{\cal K}]
Reduced composite symmetry of the domain pair (1, 2) [{\cal K}_{1,2}^*], [ {\cal K}^*]
Extended composite symmetry of a twinned crystal with a pseudo n-fold twin axis [{\cal K}(n)]

to end of page
to top of page