Index

A crystallographic point groups, Kerr Axial vector, crystallographic point groups, tensor, 130–131 61–62 crystals electrooptic coefficients, 127 C3v symmetry crystal electric field, 129 B D2d symmetry electrooptic tensor, 128 Basis functions, 31 first-order electrooptic effect/Pockels Birefringence, 114–118 effect, 123–124 Bloch wave function, 216–218 Kerr coefficients, 129 Bravais lattices, 6 Kerr effect, 124–125 Brillouin zones, 21–23 optical activity definition, 118 C gyration tensor, 120 Cartesian coordinates transformation, 28–29 gyration tensor crystallographic point Character table groups, 122–123 32 crystallographic point groups, 32–38 Neumann’s principle, 122 Pauli spin operators, 39–40 rotatory dispersion effect, 121 Coordinates transformation matrix, 29 rotatory power, 119 Crystal field symmetry spatial dispersion, 119 medium crystal field case, 86 symmetry properties, gyration tensor octahedral coordination, ligands, 86–87 components, 122 perturbation theory, 86 photoelastic effect Stark splitting, 85 elastooptical coefficients, 134 strong crystal field case, 86 photoelastic tensor-tensor form weak crystal field case, 85 difference, 133 Crystals optical properties and symmetry piezooptical coefficients, 131 birefringence refractive index coefficients, 132 crystal dielectric property, 114–115 polarization, tensor treatment double refraction, 115 electromagnetic light wave, electric indicatrix, 116–117 field, 106 linear optics fundamental equation, Jones matrices, 112 115–116 Jones vector, 110–111 optically anisotropic material left-circularly polarized light, 107 properties, 118 light wave electric field, 106–107 uniaxial crystal, wave surface, 118 Mueller matrices, 113, 114 electrooptical effect polarization transformation matrices, crystallographic point groups, 113 electrooptic tensor, 126 polarization vectors, 109

R.C. Powell, Symmetry, Group Theory, and the Physical Properties of Crystals, 225 Lecture Notes in Physics 824, DOI 10.1007/978-1-4419-7598-0, # Springer Science+Business Media, LLC 2010 226 Index polarization (cont.) four-fold rotation symmetry, 64 polarized light, 108 nonzero elements, conductivity tensor, right-circularly polarized light, 107 65 state of, 105–106 quadratic equation, coordinate system, Stokes parameters, 107–108 66 Crystal structure temperature-induced deformation, axes type, 9 67–68 Bravais lattices, 6 Taylor series, 56 crystal lattice types, 5 third-rank matter tensors cubic system, 10 crystallographic point groups, 72–73 3D crystal lattice, 6–8 mirror reflection matrix, 71 hexagonal system, 10 piezoelectric effect, 68 monoclinic system, 9 piezoelectric tensor, 69–70 noncubic crystallographic point groups, stress sensor components, 68–69 11–13 symmetry transformation properties, 70 O and Oh crystallographic point groups, 14 orthorhombic system, 9 D primitive unit cell, 4 d-electrons space groups, 12 applications, 95–96 square lattice, 4–5 Cr3þ ion electric dipole transitions, 99 stereograms, 10 Cr3þ ion energy levels, 99 symmorphic space group, 5 crystal field energy determination, 99–100 tetragonal system, 9 Oh crystal field optical transitions, 96 tetrahedral symmetry, 14–15 Oh symmetry group, 97 triclinic and trigonal systems, 9 O symmetry free ions, 98 Crystal tensor properties first-rank matter tensors E axial vector, crystallographic point Elastic compliance, 74 groups, 61–62 Electron energy bands crystallographic point groups, 59 Bloch wave function, 216–218 crystal spatial rotation effect, 57–58 cyclic groups, concepts, 215 ferroelectric effect, 60 energy vs. wave vector, 219 ferromagnetic effect, 61 Electrooptical effect nonferroelectric-ferroelectric crystals crystallographic point groups correlation, 60 electrooptic tensor, 126 pyroelectric effect, 57 Kerr tensor, 130–131 pyroelectric-ferroelectric phenomenas crystals electrooptic coefficients, 127 difference, 60 C3v symmetry crystal electric field, 129 pyroelectric tensor, 58–59 D2d symmetry electrooptic tensor, 128 scalar cause, vector effect, 57 first-order electrooptic effect/Pockels fourth-rank matter tensors effect, 123–124 crystallographic point groups, 76–77 Kerr coefficients, 129 elastic compliance, 74 Kerr effect, 124–125 elastic compliance components, 75 Hooks’s law, 74 F Neumann’s principle, 55 f-electrons second-rank matter tensors Ckq spherical harmonics, 101–102 cause and effects, 62 Judd–Ofelt theory, 103–104 conductivity tensor, 63 Nd3þ energy levels, 103 crystallographic point groups, 66 Nd3þ selection rule, 103 cubic point group symmetry operation, point group D2d character table, 102 65 rare earth ions, 100 electrical conductivity, 63 spin–orbit coupling splits, 101 Index 227

First-order electrooptic effect, 123–124 Judd–Ofelt theory, 103–104 First-rank matter tensors axial vector, crystallographic point groups, K 61–62 Kerr coefficients, 129 crystallographic point groups, 59 Kerr effect, 124–125 crystal spatial rotation effect, 57–58 Kleinman symmetry, 147 ferroelectric effect, 60 ferromagnetic effect, 61 L nonferroelectric- ferroelectric crystals Lattice vibrations correlation, 60 acoustic mode, 173 pyroelectric effect, 57 Brillouin zone symmetry operation, 177 pyroelectric-ferroelectric phenomena Oh and C4v point groups, 179 differences, 60 optic mode, 173 pyroelectric tensor, 58–59 phonon dispersion, 180 scalar cause, vector effect, 57 SrTiO3 Brillouin zone, 174–176, 178–179 Fourth-rank matter tensors strontium titanate (SrTiO3) crystal crystallographic point groups, 76–77 structure, 173–174 elastic compliance, 74 Legendre polynomials, 222–223 elastic compliance components, 75 Light wave electric field, 106–107 Hooks’s law, 74 M G Maxwell’s wave equation, 141 Group theory Mueller matrices, 113, 114 basis functions, 31 Cartesian coordinates transformation, N 28–29 Neumann’s principle, 55 character table Nonlinear optics 32 crystallographic point groups, 32–38 birefringence, 144 Pauli spin operators, 39–40 Clausius–Mossotti relationship, 137 concept, 26 controlling parameters, phase mismatch, coordinates transformation matrix, 29 143 definition, 25 coordinate system, 144–145 order of group, 25–26 coupled wave equation, 141–142 physical properties, system, 27–28 effective nonlinear optical coefficient point group symmetry, 27 coordinate system, ordinary and properties, 26 extraordinary polarization vectors, quantum mechanics 147–149 eigenfunctions Ei,50 crystallographic point groups, 148, 149 eigenvalue En,49 fourfold rotation, z-axis, 146 rectangular symmetry, 52–53 Kleinman symmetry, 147 Schrodinger€ equation, 48 nonlinear optical tensor, 146 spatial and spin functions, 51 tensor component expression, 145–146 spin–orbit interaction, 51 frequency doubling, 138 trace, transformation matrix, 30 index matching, 150–153 Gyration tensor, 120 materials, 137 Maxwell’s wave equation, 141 H momentum conservation, 140 Hooks’s law, 74 Raman scattering, 138 second-harmonic generation, 139 J beam, 143 Jahn–Teller effect, 194–197 conversion effiiency, 142 Jones matrices, 112 photon transitions, 139 Jones vector, 110–111 polarization, 139–140 228 Index second-harmonic generation (cont.) rectangular symmetry, concept, 52 power, phase mismatch, 142 rectangular symmetry, irreducible SHG efficiency maximizing representations, 52–53 noncritical phase matching, 155 Schrodinger€ equation, 48 positive crystal,, 156–157 spatial and spin functions, 51 type I phase matching, 153–154 spin–orbit interaction, 51 uniaxial negative crystals, 155 standard approach, 138–139 R two-photon absorption Raman tensor, 188 laser source producing photons, 157 Raman scattering Oh point group character table, 38, 161 32 crystallographic point groups, Raman optical transitions, 159–160 tensor, 188–193 symmetry properties, excited states, 159 material effects and applications, 194 tool, 158–159 Raman spectroscopy, 186 uniaxial crystal propagation direction, 143, schematic picture, infrared absorption, 144 187–188 Stokes scattering, 185 O Rayleigh scattering, 186 Order of group, 25–26 Reciprocal space Brillouin zones, 21–23 P description, 15 Pauli spin operators, 39–40 hexagonal crystal structure, 23 Photoelastic effect lattice periodicity function, 22 elastooptical coefficients, 134 primitive translation vectors, 21 photoelastic tensor-tensor form difference, real and reciprocal space, equivalent 133 lattices, 22, 23 piezooptical coefficients, 131 space group list, 16–20 refractive index coefficients, 132 Rotatory dispersion effect, 121 Pockels effect, 123–124 Point group symmetry, 27 S Polarization, tensor treatment Schoenfies notation, 9, 14, 15 electromagnetic light wave, electric field, Second-rank matter tensors 106 cause and effects, 62 Jones matrices, 112 conductivity tensor, 63 Jones vector, 110–111 crystallographic point groups, 66 left-circularly polarized light, 107 cubic point group symmetry operation, 65 light wave electric field, 106–107 electrical conductivity, 63 Mueller matrices, 113, 114 four-fold rotation symmetry, 64 polarization transformation matrices, 113 nonzero elements, conductivity tensor, 65 polarization vectors, 109 quadratic equation, coordinate system, 66 polarized light, 108 temperature-induced deformation, 67–68 right-circularly polarized light, 107 Solids point defects, symmetry properties state of, 105–106 crystal field symmetry Stokes parameters, 107–108 medium crystal field case, 86 Polarizability tensor, 186 octahedral coordination, ligands, 86–87 Primitive unit cell, 4 perturbation theory, 86 Pyroelectric effect, 57 Stark splitting, 85 Pyroelectric tensor, 58–59 strong crystal field case, 86 weak crystal field case, 85 Q crystal, ions energy levels Quantum mechanics, in group theory Cartesian coordinates, 92 eigenfunctions Ei,50 crystal field matrix elements, 94 eigenvalue En,49 crystal field splitting, 95 Index 229

d-electron, energy level splitting, 90 O and Oh crystallographic point groups, double-valued representation, 89 14 forced electric dipole transitions, 91 orthorhombic system, 9 group theory, 91 primitive unit cell, 4 nonrotational symmetry elements, space groups, 12 transformation matrix, 89 square lattice, 4–5 nonzero expansion coefficients, 93 stereograms, 10 orbital angular momentum quantum, 87 symmorphic space group, 5 O symmetry crystal field, 90 tetragonal system, 9 principal symmetry axis, 88 tetrahedral symmetry, 14–15 rotation operator, 88 triclinic system, 9 secular determinant elements, 95 trigonal system, 9 specific linear combination definition, 1 determination, 92 elements, 2–3 d-electrons group theory, 3 applications, 95–96 reciprocal space Cr3þ ion electric dipole transitions, 99 Brillouin zones, 21–23 Cr3þ ion energy levels, 99 description, 15 crystal field energy determination, hexagonal crystal structure, 23 99–100 lattice periodicity function, 22 Oh crystal field optical transitions, 96 primitive translation vectors, 21 Oh symmetry group, 97 real and reciprocal space, equivalent O symmetry free ions, 98 lattices, 22, 23 f-electrons space group list, 16–20 Ckq spherical harmonics, 101–102 square array, elements, 3 Judd–Ofelt theory, 103–104 Space group list, 16–20 Nd3þ energy levels, 103 Stokes parameters, 107–108 3þ Nd selection rule, 103 Strontium titanate (SrTiO3) point group D2d character table, 102 Brillouim zone, 174–176, 178–179 rare earth ions, 100 crystal structure, 173–174 spin–orbit coupling splits, 101 Symmetry and electron energy levels free ions energy level electron energy bands angular momentum operators, 81 Bloch wave function, 216–218 Coulomb interactions, 79 cyclic groups, concepts, 215 electron–electron interaction, 83 energy vz wave vector, 219 Hamiltonian represention, 80 molecular bonds spectroscopic notation, 82 Cartesian and spherical coordinates spherical harmonic functions, 83–85 relationship, 201–202 spin–orbit coupling constant, 82 D3h symmetry, AB3 molecule, 206–208 spin–orbit interaction, 82–83 D3h symmetry group character table, Solid symmetry 205–206 concept, 2 dyz orbital, p bonding, 211 crystal structure hybrid orbitals, 203 axes type, 9 molecular orbitals, 208–209 Bravais lattices, 6 Oh symmetry orbitals molecules, crystal lattice types, 5 209–211 cubic system, 10 s, p, and d electron orbitals, angular 3D crystal lattice, 6–8 factors, 202 hexagonal system, 10 spatial distribution, s, p, and d electron monoclinic system, 9 orbitals, 202–203 noncubic crystallographic point groups, Td point group character table, 11–13 204–205 230 Index molecular bonds (cont.) 32 crystallographic point groups, tetrahedral molecule Td symmetry, Raman tensor, 188–193 203–204 material effects and applications, 194 space groups character tables, 212–214 polarizability tensor, 186 symmetry property, electron energy bands Raman spectroscopy, 186 Brillouin zone, delta point, 220 Raman tensor, 188 Brillouin zone octahedral crystal Rayleigh scattering, 186 symmetry, 221 schematic picture, infrared absorption, group, wave vector, 220 Raman scattering, 187–188 Legendre polynomials, 222–223 Stokes scattering, 185 many-body problem, 223 Symmetry and lattice vibrations T Jahn–Teller effect, 194–197 Taylor series, 56 lattice vibration mode Temperature-induced deformation, 67–68 acoustic mode, 173 Third-rank matter tensors Brillouin zone symmetry operation, 177 crystallographic point groups, 72–73 Oh and C4v point groups, 179 mirror reflection matrix, 71 optic mode, 173 piezoelectric effect, 68 phonon dispersion, 180 piezoelectric tensor, 69–70 SrTiO3 Brillouin zone, 174–176, stress sensor components, 68–69 178–179 symmetry transformation properties, 70 strontium titanate (SrTiO3) crystal Two-photon absorption structure, 173–174 laser source producing photons, 157 local mode vibrations Oh point group character table, 38, 161 breathing mode, 171 optical transitions, 159–160 definition, 166 symmetry properties, excited states, 159 normal vibration mode, 168–170 tool, 158–159 octahedral configuration, 166 U Oh character table for point group, 166, 167 Uniaxial negative crystals, 155 Oh symmetry r1 coordinates transformation, 170, 171 V y15 coordinate-Oh symmetry Vibrational energy level transitions transformation, 171, 172 infrared transitions, 183–185 symmetry coordinates, 168 radiationless processes, 180 phonons, 165 radiationless transisitions, 181–183 Raman scattering