Contents -- Preface -- 1 . Batch Melting -- 1.1 Overview of Melting Models -- 1.2 Modal Batch Melting -- 1.3 Nonmodal (Eutectic) Batch Melting -- 1.3.1 Eutectic batch melting with constant Ki -- 1.3.2 Eutectic batch melting with linear change of Ki -- 1.3.3 Eutectic batch melting with linear change of Ki and linear change of pi -- 1.4 Incongruent Batch Melting -- 1.4.1 Incongruent batch melting with constant Ki -- 1.4.2 Incongruent batch melting with linear change of Ki -- 1.4.3 Incongruent batch melting with linear change of Ki and linear change of qi -- 1.5 Summary -- References -- 2 . Fractional Melting -- 2.1 Modal Fractional Melting with Constant D -- 2.2 Nonmodal (Eutectic) Fractional Melting -- 2.2.1 Eutectic fractional melting with constant Ki -- 2.2.2 Eutectic fractional melting with linear Ki -- 2.3 Incongruent Fractional Melting -- 2.3.1 Incongruent fractional melting with constant Ki -- 2.3.2 Incongruent fractional melting with linear Ki -- 2.3.3 Incongruent fractional melting with linear Ki and linear qi -- 2.4 Summary of Equations -- Appendix 2A Runga-Kutta Method -- References -- 3 . Dynamic Melting -- 3.1 Modal Dynamic Melting with Constant D -- 3.2 Eutectic Dynamic Melting -- 3.2.1 Eutectic dynamic melting with constant Ki -- 3.2.2 Eutectic dynamic melting with linear change of Ki -- 3.3 Incongruent Dynamic Melting -- 3.3.1 Incongruent dynamic melting with constant Ki -- 3.3.2 Incongruent dynamic melting with linear change of Ki -- 3.3.3 Incongruent dynamic melting with linear Ki and linear qi -- 3.4 Summary of Equations -- References -- 4 . Open-system Melting -- 4.1 Open-system Batch Melting (OBM) -- 4.1.1 OBM with one-time instantaneous addition of melt -- 4.1.2 OBM with continuous addition of melt -- 4.1.3 OBM with continuous addition of fluids and solid -- 4.2 Open-system Dynamic Melting (ODM).

4.2.1 ODM with continuous addition of melt -- 4.2.2 ODM with continuous addition of melt and solid -- 4.3 Summary of Equations -- References -- 5 . Uranium-series Disequilibrium Modeling -- 5.1 Single-stage Radioisotope Decay -- 5.2 Decay Series -- 5.2.1 Bateman equation -- 5.2.2 Secular equilibrium -- 5.3 U-series Disequilibrium Produced by Dynamic Melting -- 5.4 Forward Modeling Results -- 5.5 Discussions -- 5.5.1 The Meanings of melting rate and melt extraction rate -- 5.5.2 U-series modeling and the degree of partial melting -- 5.5.3 Activity ratios in the melt when f < before melt extraction starts -- 5.5.4 U-series modeling versus trace element modeling -- 5.6 Chromatographic Melt Transport -- 5.7 Summary of Equations -- References -- 6 . Crystallization, Assimilation, Mixing -- 6.1 Crystallization -- 6.1.1 Fractional crystallization -- 6.1.1.1 Constant D -- 6.1.1.2 Linear D -- 6.1.2 Equilibrium crystallization -- 6.2 Assimilation-fractional Crystallization -- 6.3 Two-component Mixing -- 6.3.1 Derivation -- 6.3.2 Analysis of the two-component mixing equation -- 6.4 Summary of Equations -- References -- 7 . Inverse Geochemical Modeling -- 7.1 Batch Melting Inversion of Melt Compositions -- 7.1.1 Concentration ratio method -- 7.1.2 Linear regression method -- 7.2 Dynamic Melting Inversion of Melt Compositions -- 7.3 U-Th Disequilibrium Inversion of Melt Compositions -- 7.4 Batch Melting Inversion of Whole-rock Residues -- 7.4.1 Concentration ratio method -- 7.4.2 Linear regression method -- 7.5 Batch Melting Inversion of Residual Clinopyroxenes -- 7.5.1 Concentration ratio method -- 7.5.2 Linear regression method -- Appendix 7A Newton-Raphson Method for Two Unknowns -- References -- 8 . Error Analysis -- 8.1 Random Errors -- 8.2 Statistical Treatment of Random Uncertainties -- 8.2.1 Mean and standard deviation -- 8.2.2 Weighted averages.

8.2.3 Variance, covariance and correlated errors -- 8.3 Probability Distributions -- 8.3.1 Norma/Gauss distribution -- 8.3.2 Poisson distribution -- 8.3.3 Binomial distribution -- 8.4 Systematic Errors -- 8.5 Summary -- References -- 9 . Linear Least Square Fitting -- 9.1 Least Squares: No Errors in Xi -- Yi Subject to Equal Errors -- 9.2 Least Squares: No Errors in Xi -- Yi Subject to Weighed Errors -- 9.3 Both Xi and Yi are Subject to Weighed Errors -- 9.4 Both Xi and Yi are Subject to Weighed Correlated Errors -- 9.4.1 Formulation -- 9.4.2 Error analysis -- 9.5 Summary of Linear Least Square Fitting of the Line y = a + bx -- References -- 10 . Mass Fractionation in Ionization Processes -- 10.1 Mass Fractionation Laws -- 10.1.1 Linear law -- 10.1.2 Power law -- 10.1.3 Exponential law -- 10.2 Internal and External Correction of Mass Fractionation -- 10.2.1 Internal correction -- 10.2.2 External correction -- 10.3 Dynamic Measurements of Nd and Sr Isotopic Compositions -- 10.3.1 Sr double collector analysis -- 10.3.2 Nd double collector analysis -- 10.3.3 Nd triple collector analysis -- References -- 11 . Isotope Dilution -- 11.1 Single Spike Method -- 11.1.1 Single spike equation using isotope abundances -- 11.1.2 Single spike equation using isotope ratios -- I 1.1.3 Optimization -- 1 1.2 Double Spike Method -- 1 1.3 Two Double Spikes -- 11.4 Summary of Equations -- References -- 12 . Pb Isotope Modeling -- 12.1 Why is the Tera-Wasserburg Concordia Diagram Concave Upward? -- 12.2 Why is the Conventional Concordia Plot Concave Down? -- 12.3 Why is the Holms-Houtermans Model Concave Down? -- 12.4 Why is the Two-stage Model of Pb Evolution also Concave Down? -- 12.5 Why is the U-Th-Pa Concordia Diagram Concave Down Everywhere? -- 12.6 Some Calculations Related to Pb Isotope Geochemistry -- 12.6.1 What is Concordia age?.

12.6.2 Error propagation related to Pb isotopes -- References -- 13 . Geochemical Kinematics and Dynamics -- 13.1 Diffusion -- 13.1.1 Formulation -- 13.1.2 Variation of diffusion coefficient -- 13.2 Advection and Percolation -- 13.3 Bubble Growth by Mass Transfer of Volatiles -- 13.3.1 Formulation -- 13.3.2 Quasi-static approximation -- 13.4 Bubble Growth by Decompression -- 13.4.1 Thin shell of over-pressured viscous melt -- 13.4.1.1 Instantaneous decompression -- 13.4.1.2 Linear decompression -- 13.4.2 Thin shell melt model during instantaneous decompression -- 13.5 The Trajectories of a Volcanic Bomb -- 13.5.1 No drag -- 13.5.2 Linear drag -- Appendix 13A Error Function -- References.