Contents1 Energy Level of Free Ions 11.1 Energy Levels of the Single Electron in Atoms (Free Ions) 11.2 General Properties of Energy Level in Multi-electron of Free Ions 71.3 Energy Levels of Free Transition-Metal Ions 111.4 Energy Levels of Free Rare Earth Ions 151.5 Theory of Interactions in Rare Earth Ions 24References 292 Group Theory and Quantum Theory 312.1 Mathematical Description of the Symmetry 312.2 Basic Conception of the Group 332.3 Theory of Group Representations 362.4 Direct Product Group and Direct Product Representation 402.5 Sketches of the Group in Spectroscopy 412.5.1 Finite Group 412.5.2 Permutation Group 432.5.3 Continuous Groups 462.6 Point Group and Their Representation 482.7 Symmetry and Quantum Theory of the Ions in Solids 522.8 Full Rotation Group and Angular Momentum Theory 552.9 Irreducible Tensor Operators and the Calculation of Matrix Elements 61References 673 Rare Earth Ions in Materials 693.1 Crystal Field on the Active Ions 693.2 Energy Level Splitting of the Rare Earth Ions 723.3 Crystal Field Quantum Number 813.4 Group Chain Scheme Method in Crystal Field Analysis 90References 1014 Theory of Radiative Transition 1034.1 Interactions Between Active Ions and Radiation 1034.2 Probability of Emission and Absorption Processes 1074.3 Selection Rules for Radiative Transition 1154.3.1 Selection Rules for Radiative Transition of Free Ions and Atoms 1154.3.2 Selection Rules for Radiative Transition of Ions in Materials 116References 1235 Spectroscopic Parameter and Their Calculation 1255.1 Absorption Coefficient�� Absorption (Emission) Cross-Section�� and Oscillator Strength 1255.2 Analysis of the Absorption Coefficients of Anisotropic Crystal 1325.3 Judd�COfelt Approximation and Related Parameter 1365.4 Spectroscopic Parameter Calculation of Rare Earth Ion in Crystal 1455.5 Hypersensitive Transitions 156References 1586 Phonon and Spectral Line 1616.1 Quantization of Lattice Vibration��Phonon 1616.2 Phonon Emission and Absorption in the Optical Transition 1706.3 Main Mechanisms of the Thermal Spectral Line Broadening and Shifting 1816.4 The Contribution of Single-Phonon Absorption (Emission) to the Spectral Linewidth 1836.5 The Contribution of Phonon Raman Scattering to the Spectral Linewidth 1876.6 Calculation of the Thermal Shifting of Spectral Lines 1926.7 Examples for the Calculation of Thermal Spectral Line Broadening and Shifting 196References 2017 Energy Levels and Spectroscopic Properties of Transition Metal Ions 2037.1 Energy Levels and Spectral Properties of 3d1 Electron System 2047.2 Energy Levels and Spectral Properties of 3d2 Electron System 2107.3 Energy Levels and Spectral Properties of 3d3 Electronic System 2197.4 Relative Intensity Analysis of R Line in Ruby Polarized Absorption Spectrum 2287.5 Estimation of Trivalent Chromium Ion Spectral Parameters in Solid-State Laser Materials 232References 2388 Non-radiative Transition Inside Ions 2418.1 Introduction of Non-radiative Transition Matrix Elements 2428.2 Promoting Mode and Accepting Mode in Non-radiative Transition Process 2468.3 Non-radiative Transition Probability for Weak Coupling Systems 2488.4 Parallelism Between Non-radiative Transition Probability and Radiative Transition Probability 2548.5 Temperature Dependence of Non-radiative Transition Probability in Weak Coupling Systems 2568.5.1 Experimental 2568.6 Non-radiative Transition in Strong Coupling Systems 2588.7 Nonlinear Theory of Non-radiative Transition 2658.8 Stimulated Non-radiative Transition 268References 2749 Energy Transfer and Migration Between Ions 2779.1 Theory of Resonant Energy Transfer 2789.2 Phonon-Assisted Energy Transfer Between Ions 2829.3 Statistical Theory of Energy Transfer Between Ions 2879.4 Energy Migration Between Ions 2909.5 Characteristics of Concentration Dependent Fluorescence Quenching for Self-activated Laser Crystals 303References 30610 Laser and Physical Properties of Materials 30910.1 Brief Introduction of Solid-State Laser Principle 30910.2 Quality Factor of Solid-State Laser Materials 31610.3 Relationship Between Laser Threshold and Chemical Composition of Host Materials 31810.4 Thermo-Mechanical and Thermo-Optical Propertiesof Solid-State Laser Materials 32210.5 Laser Damage and Nonlinear Optical Properties 337References 34211 Nonlinear Optical Properties of Laser Crystals and Their Applications 34511.1 Second-Order Nonlinear Optical Effect of Crystal 34711.2 Relationship Between Fundamental and Second Harmonic Waves in SFD Laser Crystal 35411.3 Nonlinear Optical Coupling Equation of SFD Laser 35911.4 Self Sum-Frequency Mixing Effect in Nonlinear Laser Crystal 36611.5 Stimula