Snow Crystals - Kenneth G. Libbrecht - Published 2019, by Kenneth G. Libbrecht I wrote this book for purposes of research, scholarship, and education only. Regarding all materials herein that I did not create, I do not intend to derive any income from these materials, nor will I authorize their use in any derivative publications, products, or other activities. By my understanding, this means that the fair-use doctrine in copyright law allows me to reproduce published material without express permission from the copyright holder (because I am using the materials in a transformative manner that will not divert any income from its creators). To the best of my abilities, I have fully cited all images or other materials that I did not create. In most cases, these materials were already freely available on the internet. You are welcome to reproduce (without express permission) any materials in this book that I created in any non-commercial pursuit of research, scholarship, or education. If you are interested in any images or other materials for commercial or other purposes, please contact me at [email protected]. Snow Crystals Contents .............................................................................................................................. 2 Forward ............................................................................................................................. 11 1. Snow Crystal Science ................................................... 15 1.1 Complex Symmetry ......................................................................................................... 17 From Clouds to Crystals ................................................................................................ 18 Faceting and Branching .................................................................................................. 20 1.2 A Brief History of Snow-Crystal Science ..................................................... 23 Early Observations ......................................................................................................... 24 Emerging Science ............................................................................................................ 25 Microscopic Snowflakes ................................................................................................. 26 Snowflake Photography ................................................................................................. 26 Crystallography ................................................................................................................ 28 Attachment Kinetics ....................................................................................................... 29 The Morphology Diagram ............................................................................................. 30 Crystal Dendrites ............................................................................................................. 32 Quantitative Snow Crystals ............................................................................................. 35 1.3 Twenty-First Century Snowflakes ................................................................. 35 Reductionism to Holism ................................................................................................ 36 The Road Ahead ............................................................................................................. 37 1.4 Why Study Snowflakes? .................................................................................. 40 1.5 No Two Alike? ................................................................................................. 42 2. Ice Crystal Structure ................................................... 45 2.1 The Phase Diagram ......................................................................................... 47 2.2 Ice Crystallography .......................................................................................... 48 Hexagonal and Cubic Ice ............................................................................................... 49 Lattice Projections .......................................................................................................... 51 Terrace Steps ................................................................................................................... 53 2 2.3 Surface Premelting ........................................................................................... 54 2.4 Ice Energetics ................................................................................................... 56 Bulk Energies ................................................................................................................... 56 Vapor Pressure and Related Quantities ....................................................................... 58 Surface Energies .............................................................................................................. 58 Terrace Step Energies ..................................................................................................... 59 A Correspondence near 0 C .......................................................................................... 61 Surface Relaxation ........................................................................................................... 61 2.5 Gibbs-Thomson Effect .................................................................................. 63 2.6 Equilibrium Shape ........................................................................................... 64 Approach to Equilibrium ............................................................................................... 65 2.7 Twinning ........................................................................................................... 66 Columnar Twins .............................................................................................................. 66 Arrowhead Twins ............................................................................................................ 68 Crossed Plates .................................................................................................................. 72 3. Attachment Kinetics ..................................................... 75 3.1 Ice Kinetics ....................................................................................................... 77 Molecular Processes ........................................................................................................ 78 Surface Characteristics ................................................................................................... 79 Facet Formation .............................................................................................................. 80 3.2 Nucleation-Limited Attachment Kinetics .................................................... 81 2D Nucleation Theory ................................................................................................... 81 Measured Facet Kinetics ................................................................................................ 83 3.3 Basal Facet Growth ......................................................................................... 84 Nucleation-Limited Growth .......................................................................................... 84 Terrace Step Energies ..................................................................................................... 85 Correspondence with Ice Growth from Liquid Water ................................................ 86 A Comprehensive Model of Basal Attachment Kinetics .......................................... 87 3.4 Prism Facet Growth ........................................................................................ 88 Attachment Kinetics on Ideal Facet Surfaces ............................................................. 89 Prism Terrace Step Energies ......................................................................................... 90 Pressure-Dependent Attachment Kinetics .................................................................. 91 A Terrace-Erosion Model .............................................................................................. 92 Structure Dependent Attachment Kinetics ................................................................. 93 The Edge-Sharpening Instability .................................................................................. 96 Inter-Facet Transport ..................................................................................................... 98 3.5 Dislocation-Mediated Growth ....................................................................... 98 3.6 Chemically Modified Attachment Kinetics ................................................ 100 3.7 A Comprehensive Model of Snow Crystal Attachment Kinetics .......... 101 Prior Attempts ............................................................................................................... 101 Ideal Facet Surfaces ...................................................................................................... 103 Pressure-Dependent Attachment Kinetics ................................................................ 103 Structure-Dependent Attachment Kinetics .............................................................. 105 The Snow Crystal Morphology Diagram ................................................................... 105 3 Next Generation Morphology Diagrams .................................................................