## Angular momentum theory applied to interactions in solidsThis book discusses those concepts of group theory that are applied to the spectra of impurity ion in crystals. Beginning with the simple hydrogen atom, spectroscopic notation and angular momentum operators are discussed. This is followed by a general discussion of angular momentum theory including Clebsch-Gordan coefficients, the Wigner-Eckart theorem, unit spherical tensors, and Racah coefficients. The extension of these techniques to the electronic configuration, nlN, for N equivalent electrons is discussed. The theory of point groups as applied to ions in solids is introduced, along with the use of the International Tables of Crystallography and character tables. The phenomenological theory of crystal fields is discussed in some detail along with the so-called free-ion parameters characterizing the Coulomb interaction, the spin-orbit interaction, and the interconfiguration interaction. The use of tables of 3-j and 6-j symbols used in the calculation of the matrix elements of the various interactions is presented, along with other tables and aids in the computation of the energy levels. The point ion model of crystal-field interaction is discussed with particular emphasis on recent work done on its development. The earlier work was applied to triply ionized rare-earth ions (4fN electronic configurations), and extensions of the point ion model are applied to the transition metal ions with the electronic configurations ndN (n=3, 4, and 5). |

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### Contents

CLEBSCHGORDAN COEFFICIENTS | 19 |

UNIT SPHERICAL TENSORS | 35 |

RACAH ALGEBRA | 49 |

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### Common terms and phrases

6-j Symbols Atomic Bibliography and References C-G coefficients C. A. Morrison Calcium Tungstate calculated Cambridge CaWOj Chem commutation consider Crystal Field Crystal Field Theory crystal-field interaction crystal-field parameters dipole discussion ELECTRONIC CONFIGURATION energy levels experimental data Faucher Field Parameters free ion given in equation given in table gives Hamiltonian Harry Diamond Laboratories Hartree-Fock International Tables IONIZED RARE-EARTH IONS J. K. Wooten Judd Karayianis lattice sums Ligand Field Theory LiYFj McGraw-Hill multipolar multipole Nd3+ Nielson and Koster notation obtain Optical Spectrum orbital Phys point charge point groups polarizability problem R. P. Leavitt Racah coefficients recouple reduced matrix reduced matrix elements relation result given Rose Rotenberg scheelite Slater integrals solid Spectra Spectroscopic Properties spherical harmonics spherical tensors spin spin-orbit interaction symmetry operations Terbium total angular momentum transition-metal ions Trivalent unit cell values vector wave functions Wigner-Eckart theorem Wiley Wortman Wybourne York